Engineering Mathematics

In memory of Elizabeth

Engineering Mathematics Sixth edition John Bird, BSc (Hons), CEng, CSci, CMath, FIET, MIEE, FIIE, FIMA, FCollT

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Newnes is an imprint of Elsevier

Newnes is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 1989 Second edition 1996 Reprinted 1998 (twice), 1999 Third edition 2001 Fourth edition 2003 Reprinted 2004 Fifth edition 2007 Sixth edition 2010 Copyright © 2001, 2003, 2007, 2010, John Bird. Published by Elsevier Ltd. All rights reserved. The right of John Bird to be identiﬁed as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [emailprotected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 978-0-08-096562-8

For information on all Newnes publications visit our Web site at www.elsevierdirect.com Typeset by: diacriTech, India Printed and bound in Hong Kong, China 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1

Contents Preface

Section 1 Number and Algebra 1

2

3

4

xii

1

Revision of fractions, decimals and percentages 1.1 Fractions 1.2 Ratio and proportion 1.3 Decimals 1.4 Percentages

3 3 5 6 9

Indices, standard form and engineering notation 2.1 Indices 2.2 Worked problems on indices 2.3 Further worked problems on indices 2.4 Standard form 2.5 Worked problems on standard form 2.6 Further worked problems on standard form 2.7 Engineering notation and common preﬁxes

11 11 12 13 15 15 16 17

Binary, octal and hexadecimal 3.1 Introduction 3.2 Binary numbers 3.3 Octal numbers 3.4 Hexadecimal numbers

19 19 19 22 24

Calculations and evaluation of formulae 4.1 Errors and approximations 4.2 Use of calculator 4.3 Conversion tables and charts 4.4 Evaluation of formulae

28 28 30 32 33

Revision Test 1

38

5 Algebra 5.1 Basic operations 5.2 Laws of indices 5.3 Brackets and factorisation 5.4 Fundamental laws and precedence 5.5 Direct and inverse proportionality

39 39 41 43 45 47

6

49 49 51 53

Further algebra 6.1 Polynomial division 6.2 The factor theorem 6.3 The remainder theorem

7

8

Partial fractions 7.1 Introduction to partial fractions 7.2 Worked problems on partial fractions with linear factors 7.3 Worked problems on partial fractions with repeated linear factors 7.4 Worked problems on partial fractions with quadratic factors

55 55

Solving simple equations 8.1 Expressions, equations and identities 8.2 Worked problems on simple equations 8.3 Further worked problems on simple equations 8.4 Practical problems involving simple equations 8.5 Further practical problems involving simple equations

61 61 61

Revision Test 2 9

55 58 59

63 65 66 68

Solving simultaneous equations 9.1 Introduction to simultaneous equations 9.2 Worked problems on simultaneous equations in two unknowns 9.3 Further worked problems on simultaneous equations 9.4 More difﬁcult worked problems on simultaneous equations 9.5 Practical problems involving simultaneous equations

69 69

10 Transposition of formulae 10.1 Introduction to transposition of formulae 10.2 Worked problems on transposition of formulae 10.3 Further worked problems on transposition of formulae 10.4 Harder worked problems on transposition of formulae

78 78

11 Solving quadratic equations 11.1 Introduction to quadratic equations 11.2 Solution of quadratic equations by factorisation 11.3 Solution of quadratic equations by ‘completing the square’ 11.4 Solution of quadratic equations by formula

84 84

69 71 73 74

78 79 81

84 86 88

vi Contents 11.5 Practical problems involving quadratic equations 11.6 The solution of linear and quadratic equations simultaneously 12 Inequalities 12.1 Introduction to inequalities 12.2 Simple inequalities 12.3 Inequalities involving a modulus 12.4 Inequalities involving quotients 12.5 Inequalities involving square functions 12.6 Quadratic inequalities 13 Logarithms 13.1 Introduction to logarithms 13.2 Laws of logarithms 13.3 Indicial equations 13.4 Graphs of logarithmic functions Revision Test 3

Multiple choice questions on Chapters 1–17

91 92 92 92 93 94 95 96 98 98 100 102 103 105

14 Exponential functions 14.1 Introduction to exponential functions 14.2 The power series for e x 14.3 Graphs of exponential functions 14.4 Napierian logarithms 14.5 Laws of growth and decay

106 106 107 109 111 113

15 Number sequences 15.1 Arithmetic progressions 15.2 Worked problems on arithmetic progressions 15.3 Further worked problems on arithmetic progressions 15.4 Geometric progressions 15.5 Worked problems on geometric progressions 15.6 Further worked problems on geometric progressions 15.7 Combinations and permutations

117 117

16 The binomial series 16.1 Pascal’s triangle 16.2 The binomial series 16.3 Worked problems on the binomial series 16.4 Further worked problems on the binomial series 16.5 Practical problems involving the binomial theorem

125 125 126 126

17 Solving equations by iterative methods 17.1 Introduction to iterative methods 17.2 The Newton–Raphson method 17.3 Worked problems on the Newton–Raphson method

134 134 134

Revision Test 4

138

89

117 118 120 121 122 123

128 131

Section 2 Areas and Volumes

143

18 Areas of common shapes 18.1 Introduction 18.2 Properties of quadrilaterals 18.3 Areas of common shapes 18.4 Worked problems on areas of common shapes 18.5 Further worked problems on areas of plane ﬁgures 18.6 Worked problems on areas of composite ﬁgures 18.7 Areas of similar shapes

145 145 145 146

19 The circle 19.1 Introduction 19.2 Properties of circles 19.3 Radians and degrees 19.4 Arc length and area of circles and sectors 19.5 Worked problems on arc length and area of circles and sectors 19.6 The equation of a circle

154 154 154 156 157

20 Volumes and surface areas of common solids 20.1 Introduction 20.2 Volumes and surface areas of regular solids 20.3 Worked problems on volumes and surface areas of regular solids 20.4 Further worked problems on volumes and surface areas of regular solids 20.5 Volumes and surface areas of frusta of pyramids and cones 20.6 The frustum and zone of a sphere 20.7 Prismoidal rule 20.8 Volumes of similar shapes

162 162

21 Irregular areas and volumes and mean values of waveforms 21.1 Area of irregular ﬁgures 21.2 Volumes of irregular solids 21.3 The mean or average value of a waveform Revision Test 5

Section 3 Trigonometry

146 149 151 152

157 160

162 163 165 170 173 176 178 179 179 181 182 187

189

135 137

22 Introduction to trigonometry 22.1 Trigonometry 22.2 The theorem of Pythagoras

191 191 191

vii

Contents 22.3 Trigonometric ratios of acute angles 22.4 Fractional and surd forms of trigonometric ratios 22.5 Evaluating trigonometric ratios of any angles 22.6 Solution of right-angled triangles 22.7 Angles of elevation and depression 22.8 Trigonometric approximations for small angles

192 194 195 199 201 203

23 Trigonometric waveforms 23.1 Graphs of trigonometric functions 23.2 Angles of any magnitude 23.3 The production of a sine and cosine wave 23.4 Sine and cosine curves 23.5 Sinusoidal form A sin(ωt ± α) 23.6 Waveform harmonics

204 204 204 207 207 211 214

24 Cartesian and polar co-ordinates 24.1 Introduction 24.2 Changing from Cartesian into polar co-ordinates 24.3 Changing from polar into Cartesian co-ordinates 24.4 Use of Pol/Rec functions on calculators

216 216

Revision Test 6

216 218 219 221

25 Triangles and some practical applications 25.1 Sine and cosine rules 25.2 Area of any triangle 25.3 Worked problems on the solution of triangles and their areas 25.4 Further worked problems on the solution of triangles and their areas 25.5 Practical situations involving trigonometry 25.6 Further practical situations involving trigonometry

222 222 222

26 Trigonometric identities and equations 26.1 Trigonometric identities 26.2 Worked problems on trigonometric identities 26.3 Trigonometric equations 26.4 Worked problems (i) on trigonometric equations 26.5 Worked problems (ii) on trigonometric equations 26.6 Worked problems (iii) on trigonometric equations 26.7 Worked problems (iv) on trigonometric equations

231 231

222

27 Compound angles 27.1 Compound angle formulae 27.2 Conversion of a sin ωt + b cos ωt into R sin(ωt + α) 27.3 Double angles 27.4 Changing products of sines and cosines into sums or differences 27.5 Changing sums or differences of sines and cosines into products

238 238 240 244 245 246

Revision Test 7

248

Multiple choice questions on Chapters 18–27

249

Section 4 Graphs

255

28 Straight line graphs 28.1 Introduction to graphs 28.2 The straight line graph 28.3 Practical problems involving straight line graphs

257 257 257

29 Reduction of non-linear laws to linear form 29.1 Determination of law 29.2 Determination of law involving logarithms

269 269

30 Graphs with logarithmic scales 30.1 Logarithmic scales 30.2 Graphs of the form y = ax n 30.3 Graphs of the form y = ab x 30.4 Graphs of the form y = ae kx

277 277 277 280 281

31 Graphical solution of equations 31.1 Graphical solution of simultaneous equations 31.2 Graphical solution of quadratic equations 31.3 Graphical solution of linear and quadratic equations simultaneously 31.4 Graphical solution of cubic equations

284

32 Functions and their curves 32.1 Standard curves 32.2 Simple transformations 32.3 Periodic functions 32.4 Continuous and discontinuous functions 32.5 Even and odd functions 32.6 Inverse functions

292 292 294 299 299 299 301

263

272

224 226 228

231 232 233 234

284 285 289 290

235 236

Revision Test 8

305

viii Contents Section 5 Complex Numbers

307

33 Complex numbers 33.1 Cartesian complex numbers 33.2 The Argand diagram 33.3 Addition and subtraction of complex numbers 33.4 Multiplication and division of complex numbers 33.5 Complex equations 33.6 The polar form of a complex number 33.7 Multiplication and division in polar form 33.8 Applications of complex numbers

311 313 314 316 317

34 De Moivre’s theorem 34.1 Introduction 34.2 Powers of complex numbers 34.3 Roots of complex numbers

321 321 321 322

Section 6 Vectors 35 Vectors 35.1 35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9

Introduction Scalars and vectors Drawing a vector Addition of vectors by drawing Resolving vectors into horizontal and vertical components Addition of vectors by calculation Vector subtraction Relative velocity i, j, and k notation

36 Methods of adding alternating waveforms 36.1 Combination of two periodic functions 36.2 Plotting periodic functions 36.3 Determining resultant phasors by drawing 36.4 Determining resultant phasors by the sine and cosine rules 36.5 Determining resultant phasors by horizontal and vertical components 36.6 Determining resultant phasors by complex numbers

309 309 310 310

325 327 327 327 327 328 330 331 336 338 339 341 341 341 343 344 346 348

Revision Test 9

351

Section 7 Statistics

353

37 Presentation of statistical data 37.1 Some statistical terminology 37.2 Presentation of ungrouped data 37.3 Presentation of grouped data

355 355 356 360

38 Measures of central tendency and dispersion 38.1 Measures of central tendency 38.2 Mean, median and mode for discrete data 38.3 Mean, median and mode for grouped data 38.4 Standard deviation 38.5 Quartiles, deciles and percentiles

367 367 367 368 370 372

39 Probability 39.1 Introduction to probability 39.2 Laws of probability 39.3 Worked problems on probability 39.4 Further worked problems on probability 39.5 Permutations and combinations

374 374 375 375 377 379

Revision Test 10

381

40 The binomial and Poisson distributions 40.1 The binomial distribution 40.2 The Poisson distribution

382 382 385

41 The normal distribution 41.1 Introduction to the normal distribution 41.2 Testing for a normal distribution

388 388 393

Revision Test 11

396

Multiple choice questions on Chapters 28–41

397

Section 8 Differential Calculus

403

42 Introduction to differentiation 42.1 Introduction to calculus 42.2 Functional notation 42.3 The gradient of a curve 42.4 Differentiation from ﬁrst principles 42.5 Differentiation of y = ax n by the general rule 42.6 Differentiation of sine and cosine functions 42.7 Differentiation of e ax and ln ax

405 405 405 406 407

43 Methods of differentiation 43.1 Differentiation of common functions 43.2 Differentiation of a product 43.3 Differentiation of a quotient 43.4 Function of a function 43.5 Successive differentiation

415 415 417 418 420 421

44 Some applications of differentiation 44.1 Rates of change 44.2 Velocity and acceleration 44.3 Turning points 44.4 Practical problems involving maximum and minimum values

423 423 424 427

409 411 412

431

Contents 44.5 Tangents and normals 44.6 Small changes Revision Test 12

434 435 437

45 Differentiation of parametric equations 45.1 Introduction to parametric equations 45.2 Some common parametric equations 45.3 Differentiation in parameters 45.4 Further worked problems on differentiation of parametric equations

438 438 438 439

46 Differentiation of implicit functions 46.1 Implicit functions 46.2 Differentiating implicit functions 46.3 Differentiating implicit functions containing products and quotients 46.4 Further implicit differentiation

443 443 443

47 Logarithmic differentiation 47.1 Introduction to logarithmic differentiation 47.2 Laws of logarithms 47.3 Differentiation of logarithmic functions 47.4 Differentiation of further logarithmic functions 47.5 Differentiation of [ f (x)]x

448 448 448 448

Revision Test 13

Section 9 Integral Calculus 48 Standard integration 48.1 The process of integration 48.2 The general solution of integrals of the form ax n 48.3 Standard integrals 48.4 Deﬁnite integrals

440

444 445

449 451 453

455 457 457 457 458 461

49 Integration using algebraic substitutions 49.1 Introduction 49.2 Algebraic substitutions 49.3 Worked problems on integration using algebraic substitutions 49.4 Further worked problems on integration using algebraic substitutions 49.5 Change of limits

464 464 464

50 Integration using trigonometric substitutions 50.1 Introduction 50.2 Worked problems on integration of sin2 x,cos2 x,tan 2 x and cot2 x 50.3 Worked problems on powers of sines and cosines

469 469

464 466 466

469 471

50.4 Worked problems on integration of products of sines and cosines 50.5 Worked problems on integration using the sin θ substitution 50.6 Worked problems on integration using the tan θ substitution Revision Test 14

472 473 475 476

51 Integration using partial fractions 51.1 Introduction 51.2 Worked problems on integration using partial fractions with linear factors 51.3 Worked problems on integration using partial fractions with repeated linear factors 51.4 Worked problems on integration using partial fractions with quadratic factors 52 The t = tan θ2 substitution 52.1 Introduction 52.2 Worked problems on the t = tan θ2 substitution 52.3 Further worked problems on the t = tan θ2 substitution

477 477 477 479 480 482 482 482 484

53 Integration by parts 53.1 Introduction 53.2 Worked problems on integration by parts 53.3 Further worked problems on integration by parts

486 486 486

54 Numerical integration 54.1 Introduction 54.2 The trapezoidal rule 54.3 The mid-ordinate rule 54.4 Simpson’s rule

491 491 491 493 495

Revision Test 15

488

499

55 Areas under and between curves 55.1 Area under a curve 55.2 Worked problems on the area under a curve 55.3 Further worked problems on the area under a curve 55.4 The area between curves

500 500 501

56 Mean and root mean square values 56.1 Mean or average values 56.2 Root mean square values

509 509 511

57 Volumes of solids of revolution 57.1 Introduction 57.2 Worked problems on volumes of solids of revolution 57.3 Further worked problems on volumes of solids of revolution

513 513

504 506

514 515

ix

x Contents 58 Centroids of simple shapes 58.1 Centroids 58.2 The ﬁrst moment of area 58.3 Centroid of area between a curve and the x-axis 58.4 Centroid of area between a curve and the y-axis 58.5 Worked problems on centroids of simple shapes 58.6 Further worked problems on centroids of simple shapes 58.7 Theorem of Pappus 59 Second moments of area 59.1 Second moments of area and radius of gyration 59.2 Second moment of area of regular sections 59.3 Parallel axis theorem 59.4 Perpendicular axis theorem 59.5 Summary of derived results 59.6 Worked problems on second moments of area of regular sections 59.7 Worked problems on second moments of area of composite areas Revision Test 16

61.6 The determinant of a 3 by 3 matrix 61.7 The inverse or reciprocal of a 3 by 3 matrix

518 518 518 518 519 519 520 523 527

62 The solution of simultaneous equations by matrices and determinants 62.1 Solution of simultaneous equations by matrices 62.2 Solution of simultaneous equations by determinants 62.3 Solution of simultaneous equations using Cramers rule Revision Test 17

564 566 568 568 570 574 575

527

Section 11 Differential Equations 527 528 528 528 529 532 534

63 Introduction to differential equations 63.1 Family of curves 63.2 Differential equations 63.3 The solution of equations of the form dy = f (x) dx 63.4 The solution of equations of the form dy = f (y) dx 63.5 The solution of equations of the form dy = f (x) · f (y) dx

577 579 579 580 580

582 584

Section 10 Further Number and Algebra 535 60 Boolean algebra and logic circuits 60.1 Boolean algebra and switching circuits 60.2 Simplifying Boolean expressions 60.3 Laws and rules of Boolean algebra 60.4 De Morgan’s laws 60.5 Karnaugh maps 60.6 Logic circuits 60.7 Universal logic gates

537 537 542 542 544 545 550 554

61 The theory of matrices and determinants 61.1 Matrix notation 61.2 Addition, subtraction and multiplication of matrices 61.3 The unit matrix 61.4 The determinant of a 2 by 2 matrix 61.5 The inverse or reciprocal of a 2 by 2 matrix

558 558 558 562 562 563

Revision Test 18

587

Multiple choice questions on Chapters 42–63

588

Answers to multiple choice questions

592

Index

593

On the Website 64 Linear correlation 64.1 Introduction to linear correlation 64.2 The product-moment formula for determining the linear correlation coefﬁcient 64.3 The signiﬁcance of a coefﬁcient of correlation 64.4 Worked problems on linear correlation

1 1 1 2 2

xi

Contents 65 Linear regression 65.1 Introduction to linear regression 65.2 The least-squares regression lines 65.3 Worked problems on linear regression 66 Sampling and estimation theories 66.1 Introduction 66.2 Sampling distributions

6 6 6 7 12 12 12

66.3 The sampling distribution of the means 66.4 The estimation of population parameters based on a large sample size 66.5 Estimating the mean of a population based on a small sample size Revision Test 19

12 15 20 24

Preface Engineering Mathematics 6th Edition covers a wide range of syllabus requirements. In particular, the book is most suitable for the latest National Certiﬁcate and Diploma courses and City & Guilds syllabuses in Engineering. This text will provide a foundation in mathematical principles, which will enable students to solve mathematical, scientiﬁc and associated engineering principles. In addition, the material will provide engineering applications and mathematical principles necessary for advancement onto a range of Incorporated Engineer degree proﬁles. It is widely recognised that a students’ ability to use mathematics is a key element in determining subsequent success. First year undergraduates who need some remedial mathematics will also ﬁnd this book meets their needs. In Engineering Mathematics 6th Edition, new material is included on logarithms, exponential functions, vectors and methods of adding alternating waveforms. A feature of this sixth edition is that a free Internet download is available of a sample of solutions (some 1250) of the 1750 further problems contained in the book – see page xiii. Another new feature is a free Internet download of a PowerPoint presentation of all 500 illustrations contained in the text – see page xiii. Three statistics chapters – on linear correlation, linear regression and sampling and estimation theories – are available to all via the Internet. Throughout the text theory is introduced in each chapter by a simple outline of essential deﬁnitions, formulae, laws and procedures. The theory is kept to a minimum, for problem solving is extensively used to establish and exemplify the theory. It is intended that readers will gain real understanding through seeing problems solved and then through solving similar problems themselves. For clarity, the text is divided into 11 topic areas, these being: number and algebra, areas and volumes, trigonometry, graphs, complex numbers, vectors, statistics, differential calculus, integral calculus, further number and algebra and differential equations

This new edition covers, in particular, the following syllabuses: (i) Mathematics for Technicians, the core unit for National Certiﬁcate/Diploma courses in Engineering, to include all or part of the following chapters: 1. Algebraic methods: 2, 5, 11, 13, 14, 28, 30 (1, 4, 8, 9 and 10 for revision) 2. Trigonometric methods and areas and volumes: 18–20, 22–25, 33, 34 3. Statistical methods: 37, 38 4. Elementary calculus: 42, 48, 55 (ii) Further Mathematics for Technicians, the optional unit for National Certiﬁcate/Diploma courses in Engineering, to include all or part of the following chapters: 1. Advanced graphical techniques: 29–31 2. Algebraic techniques: 15, 35, 38 3. Trigonometry: 23, 26, 27, 34 4. Calculus: 42–44, 48, 55–56 (iii) Mathematics contents of City & Guilds Technician Certiﬁcate/Diploma and Advanced Diploma in Engineering and Telecommunication System disciplines. (iv) Any introductory/access/foundation course involving Engineering Mathematics at University, Colleges of Further and Higher education and in schools. Each topic considered in the text is presented in a way that assumes in the reader little previous knowledge of that topic. Engineering Mathematics 6th Edition provides a follow-up to Basic Engineering Mathematics 5th Edition and a lead into Higher Engineering Mathematics 6th Edition. This textbook contains over 1000 worked problems, followed by some 1750 further problems (all with answers). The further problems are contained within some 230 exercises; each exercise follows on directly

Preface from the relevant section of work, every two or three pages. In addition, the text contains 238 multiplechoice questions. Where at all possible, the problems mirror practical situations found in engineering and science. 513 line diagrams enhance the understanding of the theory. At regular intervals throughout the text are some 18 Revision Tests to check understanding. For example, Revision Test 1 covers material contained in Chapters 1 to 4, Revision Test 2 covers the material in Chapters 5 to 8, and so on. These Revision Tests do not have answers given since it is envisaged that lecturers could set the tests for students to attempt as part of their course structure. Lecturers may obtain a complimentary set of solutions of the Revision Tests in an Instructor’s Manual available from the publishers via the internet – see the next column. A list of Essential Formulae is included in the Instructor’s Manual for convenience of reference. Learning by Example is at the heart of Engineering Mathematics 6th Edition. JOHN BIRD Royal Naval School of Marine Engineering, HMS Sultan, formerly University of Portsmouth and Highbury College, Portsmouth

Free web downloads available at: www.textbooks.elsevier.com Additional material on statistics Chapters on linear correlation, linear regression and sampling and estimation theories are available to all. Sample of worked solutions to exercises Within the text are some 1750 further problems arranged within 230 exercises. A sample of over 1250 worked solutions has been prepared and is available to lecturers only. Instructor’s manual This provides full worked solutions and mark scheme for all 18 Revision Tests in this book. The material is available to lecturers only. PowerPoint presentation This provides on PowerPoint all of the 513 illustrations used in this new sixth edition. The material is available to lecturers only.

xiii

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Section 1

Number and Algebra

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Chapter 1

Revision of fractions, decimals and percentages 1.1

Alternatively:

Fractions

When 2 is divided by 3, it may be written as 23 or 2/3 or 2/3. 23 is called a fraction. The number above the line, i.e. 2, is called the numerator and the number below the line, i.e. 3, is called the denominator. When the value of the numerator is less than the value of the denominator, the fraction is called a proper fraction; thus 23 is a proper fraction. When the value of the numerator is greater than the denominator, the fraction is called an improper fraction. Thus 73 is an improper fraction and can also be expressed as a mixed number, that is, an integer and a proper fraction. Thus the improper fraction 73 is equal to the mixed number 2 13 . When a fraction is simpliﬁed by dividing the numerator and denominator by the same number, the process is called cancelling. Cancelling by 0 is not permissible. Problem 1. Simplify:

1 2 + 3 7

The lowest common multiple (i.e. LCM) of the two denominators is 3 × 7, i.e. 21. Expressing each fraction so that their denominators are 21, gives: 7 6 1 2 1 7 2 3 + = × + × = + 3 7 3 7 7 3 21 21 =

7 + 6 13 = 21 21

DOI: 10.1016/B978-0-08-096562-8.00001-8

1 2 + = 3 7

Step (2) Step (3) ↓ ↓ (7 × 1)+ (3 × 2) 21 ↑ Step (1)

Step 1: the LCM of the two denominators; Step 2: for the fraction 13 , 3 into 21 goes 7 times, 7 ×the numerator is 7 ×1; Step 3: for the fraction 27 , 7 into 21 goes 3 times, 3 ×the numerator is 3 ×2. Thus

1 2 7 +6 13 + = = as obtained previously. 3 7 21 21

2 1 Problem 2. Find the value of 3 − 2 3 6 One method is to split the mixed numbers into integers and their fractional parts. Then 1 2 1 2 − 2+ 3 −2 = 3+ 3 6 3 6 1 2 = 3+ −2− 3 6 4 1 3 1 = 1+ − = 1 = 1 6 6 6 2 Another method is to express the mixed numbers as improper fractions.

Section 1

4 Engineering Mathematics 2 9 2 11 9 Since 3 = , then 3 = + = 3 3 3 3 3 1 12 1 13 Similarly, 2 = + = 6 6 6 6 2 1 11 13 22 13 9 1 Thus 3 − 2 = − = − = = 1 3 6 3 6 6 6 6 2 as obtained previously. Problem 3.

=

8 17 24 8 8×1×8 = × × 5 13 71 5×1×1

=

4 64 = 12 5 5

Problem 6.

Simplify:

3 3 12 7 ÷ = 12 7 21 21

Determine the value of 1 2 5 4 −3 +1 8 4 5

5 1 2 5 1 2 4 − 3 + 1 = (4 − 3 + 1) + − + 8 4 5 8 4 5

Problem 4.

= 2+

5 × 5 − 10 × 1 + 8 × 2 40

= 2+

25 − 10 + 16 40

= 2+

31 31 =2 40 40

Find the value of

3 14 × 7 15

Multiplying both numerator and denominator by the reciprocal of the denominator gives: 3 7 = 12 21

7

×

1 14 1 × 14 14 = × = 155 7 5 7×5

Dividing numerator and denominator by 7 gives: 1 × 14 2 1×2 2 = = 7 × 5 1 ×5 5 1 This process of dividing both the numerator and denominator of a fraction by the same factor(s) is called cancelling. Problem 5.

1 3 3 Evaluate: 1 × 2 × 3 5 3 7

Mixed numbers must be expressed as improper fractions before multiplication can be performed. Thus, 3 1 3 1 ×2 ×3 5 3 7 6 1 21 3 5 3 + × + × + = 5 5 3 3 7 7

13

×

17 1 12 1

21

×

21 3 12 4 21 1 12 1

3 3 = 4 = 1 4

This method can be remembered by the rule: invert the second fraction and change the operation from division to multiplication. Thus: 3 12 1 3 3 21 3 = as obtained previously. ÷ = × 7 21 1 7 12 4 4

Dividing numerator and denominator by 3 gives: 13

3 12 ÷ 7 21

Problem 7.

1 3 Find the value of 5 ÷ 7 5 3

The mixed numbers must be expressed as improper fractions. Thus, 3 1 28 22 5 ÷7 = ÷ = 5 3 5 3 Problem 8.

14

42 3 28 = × 5 22 11 55

Simplify: 1 2 1 3 1 − + ÷ × 3 5 4 8 3

The order of precedence of operations for problems containing fractions is the same as that for integers, i.e. remembered by BODMAS (Brackets, Of, Division, Multiplication, Addition and Subtraction). Thus, 2 1 3 1 1 − + ÷ × 3 5 4 8 3

=

31 1 4×2+5×1 − ÷ 3 20 24 8

(B)

=

82 13 1 − × 3 5 20 1

(D)

1 26 − 3 5 (5 × 1) − (3 × 26) = 15 −73 13 = = −4 15 15

=

2. (a)

(M) (S)

7 of 6 =

3 1 7 5 41 ÷ − = × + 6 4 8 16 2 = = = = = =

5. (a)

(B)

6. (a)

35 82 1 + − 24 3 2 35 + 656 1 − 24 2 691 1 − 24 2 691 − 12 24 679 7 = 28 24 24

5 3 (b) 12 49

3 7 2 13 7 4 × ×1 (b) ×4 ×3 5 9 7 17 11 39 3 (b) 11 (a) 5 1 5 3 45 ÷ (b) 1 ÷ 2 8 64 3 9

8 12 (b) 15 23 8 1 7 1 3 + ÷ − 1 2 5 15 3 24 3 15 5 4 7 of 15 × + ÷ 5 15 7 4 16 5 1 2 1 3 2 13 − × − ÷ + 4 3 3 5 7 126 2 1 3 28 1 2 ÷ +1 2 ×1 + 3 4 3 4 5 55 (a)

(O)

7 5 41 16 2 1 × + × − 6 4 18 3 2

17 15 3 5 × (b) × 4 9 35 119

(a)

1 1 1 3 1 3 −2 +5 ÷ − 2 4 8 16 2 7 1 41 3 1 of 1 + ÷ − 6 4 8 16 2

47 43 (b) (a) 77 63

2 1 4 5 3 (b) 3 − 4 + 1 3. (a) 10 − 8 7 3 4 5 6 17 16 (b) (a) 1 21 60 4. (a)

Problem 9. Determine the value of 1 1 1 3 1 7 of 3 − 2 +5 ÷ − 6 2 4 8 16 2

2 3 2 1 2 + (b) − + 7 11 9 7 3

(D)

7.

(M)

8.

(A)

9.

(A)

10.

(S)

11. If a storage tank is holding 450 litres when it is three-quarters full, how much will it contain when it is two-thirds full? [400 litres] 12. Three people, P, Q and R contribute to a fund. P provides 3/5 of the total, Q provides 2/3 of the remainder, and R provides £8. Determine (a) the total of the fund, (b) the contributions of P and Q. [(a) £60 (b) £36, £16]

Now try the following exercise Exercise 1 Further problems on fractions Evaluate the following: 1. (a)

7 1 1 2 + (b) − 2 5 16 4

1.2 (a)

9 3 (b) 10 16

Ratio and proportion

The ratio of one quantity to another is a fraction, and is the number of times one quantity is contained in another quantity of the same kind. If one quantity is

5

Section 1

Revision of fractions, decimals and percentages

Section 1

6 Engineering Mathematics directly proportional to another, then as one quantity doubles, the other quantity also doubles. When a quantity is inversely proportional to another, then as one quantity doubles, the other quantity is halved. Problem 10. A piece of timber 273 cm long is cut into three pieces in the ratio of 3 to 7 to 11. Determine the lengths of the three pieces The total number of parts is 3+7 + 11, that is, 21. Hence 21 parts correspond to 273 cm 1 part corresponds to

273 = 13 cm 21

3 parts correspond to 3 × 13 = 39 cm

1 person takes three times as long, i.e. 4 × 3 =12 hours, 5 people can do it in one ﬁfth of the time that one 12 person takes, that is hours or 2 hours 24 minutes. 5 Now try the following exercise Exercise 2 1.

Divide 621 cm in the ratio of 3 to 7 to 13. [81 cm to 189 cm to 351 cm]

2.

When mixing a quantity of paints, dyes of four different colours are used in the ratio of 7 : 3 : 19 : 5. If the mass of the ﬁrst dye used is 3 12 g, determine the total mass of the dyes used. [17 g]

3.

Determine how much copper and how much zinc is needed to make a 99 kg brass ingot if they have to be in the proportions copper : zinc: :8 : 3 by mass. [72 kg : 27 kg]

4.

It takes 21 hours for 12 men to resurface a stretch of road. Find how many men it takes to resurface a similar stretch of road in 50 hours 24 minutes, assuming the work rate remains constant. [5]

5.

It takes 3 hours 15 minutes to ﬂy from city A to city B at a constant speed. Find how long the journey takes if

7 parts correspond to 7 × 13 = 91 cm 11 parts correspond to 11 × 13 = 143 cm i.e. the lengths of the three pieces are 39 cm, 91 cm and 143 cm. (Check: 39 + 91 +143 =273) Problem 11. A gear wheel having 80 teeth is in mesh with a 25 tooth gear. What is the gear ratio?

Gear ratio = 80 :25 =

80 16 = = 3.2 25 5

i.e. gear ratio =16 : 5 or 3.2 : 1 Problem 12. An alloy is made up of metals A and B in the ratio 2.5 : 1 by mass. How much of A has to be added to 6 kg of B to make the alloy? Ratio A : B: :2.5 : 1 (i.e. A is to B as 2.5 is to 1) or A 2.5 = = 2.5 B 1 A When B =6 kg, = 2.5 from which, 6 A =6 × 2.5 = 15 kg Problem 13. If 3 people can complete a task in 4 hours, how long will it take 5 people to complete the same task, assuming the rate of work remains constant The more the number of people, the more quickly the task is done, hence inverse proportion exists. 3 people complete the task in 4 hours.

Further problems on ratio and proportion

(a) the speed is 1 12 times that of the original speed and (b) if the speed is three-quarters of the original speed. [(a) 2 h 10 min (b) 4 h 20 min]

1.3

Decimals

The decimal system of numbers is based on the digits 0 to 9. A number such as 53.17 is called a decimal fraction, a decimal point separating the integer part, i.e. 53, from the fractional part, i.e. 0.17 A number which can be expressed exactly as a decimal fraction is called a terminating decimal and those which cannot be expressed exactly as a decimal fraction are called non-terminating decimals. Thus, 32 = 1.5

is a terminating decimal, but 43 = 1.33333. . . is a nonterminating decimal. 1.33333. . . can be written as 1.3, called ‘one point-three recurring’. The answer to a non-terminating decimal may be expressed in two ways, depending on the accuracy required: (i) correct to a number of signiﬁcant ﬁgures, that is, ﬁgures which signify something, and (ii) correct to a number of decimal places, that is, the number of ﬁgures after the decimal point. The last digit in the answer is unaltered if the next digit on the right is in the group of numbers 0, 1, 2, 3 or 4, but is increased by 1 if the next digit on the right is in the group of numbers 5, 6, 7, 8 or 9. Thus the nonterminating decimal 7.6183. . . becomes 7.62, correct to 3 signiﬁcant ﬁgures, since the next digit on the right is 8, which is in the group of numbers 5, 6, 7, 8 or 9. Also 7.6183. . . becomes 7.618, correct to 3 decimal places, since the next digit on the right is 3, which is in the group of numbers 0, 1, 2, 3 or 4. Problem 14. Evaluate: 42.7 + 3.04 +8.7 + 0.06 The numbers are written so that the decimal points are under each other. Each column is added, starting from the right. 42.7 3.04 8.7 0.06 54.50

The sum of the positive decimal fractions is 23.4 + 32.68 = 56.08 The sum of the negative decimal fractions is 17.83 + 57.6 = 75.43 Taking the sum of the negative decimal fractions from the sum of the positive decimal fractions gives: 56.08 −75.43 i.e. −(75.43 − 56.08) =−19.35 Problem 17. Determine the value of 74.3 × 3.8 When multiplying decimal fractions: (i) the numbers are multiplied as if they are integers, and (ii) the position of the decimal point in the answer is such that there are as many digits to the right of it as the sum of the digits to the right of the decimal points of the two numbers being multiplied together. Thus (i)

743 38 5 944 22 290 28 234

(ii) As there are (1 + 1) =2 digits to the right of the decimal points of the two numbers being multiplied together, (74.3× 3.8), then 74.3 × 3.8 = 282.34

Thus 42.7 + 3.04 +8.7 +0.06 = 54.50 Problem 15. Take 81.70 from 87.23 The numbers are written with the decimal points under each other. 87.23 −81.70 5.53 Thus 87.23 −81.70 =5.53

Problem 18. Evaluate 37.81 ÷1.7, correct to (i) 4 signiﬁcant ﬁgures and (ii) 4 decimal places

37.81 ÷ 1.7 =

37.81 1.7

The denominator is changed into an integer by multiplying by 10. The numerator is also multiplied by 10 to keep the fraction the same. Thus 37.81 ÷ 1.7 =

37.81 × 10 1.7 × 10

Problem 16. Find the value of 23.4 − 17.83 − 57.6 + 32.68

=

378.1 17

7

Section 1

Revision of fractions, decimals and percentages

Section 1

8 Engineering Mathematics The long division is similar to the long division of integers and the ﬁrst four steps are as shown: 22.24117..

(b) For mixed numbers, it is only necessary to convert the proper fraction part of the mixed number to a decimal fraction. Thus, dealing with the 78 gives:

17 378.100000 34 __ 38 34 __

Thus 5

41 34 __

(i) 37.81 ÷1.7 = 22.24, correct to 4 signiﬁcant ﬁgures, and (ii) 37.81 ÷1.7 = 22.2412, correct to 4 decimal places. Problem 19. Convert (a) 0.4375 to a proper fraction and (b) 4.285 to a mixed number 0.4375 ×10 000 without 10 000

4375 10 000

1.

23.6 +14.71 −18.9 −7.421

2.

73.84 −113.247 +8.21 − 0.068

3.

3.8 ×4.1 × 0.7

[10.906]

4.

374.1 ×0.006

[2.2446]

5.

421.8 ÷17, (a) correct to 4 signiﬁcant ﬁgures and (b) correct to 3 decimal places. [(a) 24.81 (b) 24.812]

Problem 20.

57 285 =4 1000 200

Express as decimal fractions: 7 9 and (b) 5 (a) 16 8

(a) To convert a proper fraction to a decimal fraction, the numerator is divided by the denominator. Division by 16 can be done by the long division method, or, more simply, by dividing by 2 and then 8: 4.50 2 9.00 Thus

7.

7 0.4375 = 16

(b) Similarly, 4.285 =4

9 = 0.5625 16

0.5625 8 4.5000

[11.989] [−31.265]

6.

875 175 35 7 4375 = = = = 10 000 2000 400 80 16

Further problems on decimals

In Problems 1 to 6, determine the values of the expressions given:

By cancelling

i.e.

7 = 0.875 8

7 = 5.875 8

Exercise 3

20

(a) 0.4375 can be written as changing its value,

i.e.

Now try the following exercise

70 68 __

i.e. 0.4375 =

0.875 8 7.000

0.0147 , (a) correct to 5 decimal places and 2.3 (b) correct to 2 signiﬁcant ﬁgures. [(a) 0.00639 (b) 0.0064] Convert to proper fractions: (a) 0.65 (b) 0.84 (c) 0.0125 (d) 0.282 and (e) 0.024 21 1 141 3 13 (b) (c) (d) (e) (a) 20 25 80 500 125

8.

Convert to mixed numbers: (a) 1.82 (b) 4.275 (c) 14.125 (d) 15.35 and (e) 16.2125 ⎤ ⎡ 11 1 41 ⎢ (a) 1 50 (b) 4 40 (c) 14 8 ⎥ ⎥ ⎢ ⎦ ⎣ 7 17 (d) 15 (e) 16 20 80 In Problems 9 to 12, express as decimal fractions to the accuracy stated: 9.

4 , correct to 5 signiﬁcant ﬁgures. 9

[0.44444]

17 , correct to 5 decimal places. 27

10.

[0.62963]

9 , correct to 4 signiﬁcant ﬁgures. 16

11.

1

12.

31 13 , correct to 2 decimal places. 37

13.

To convert fractions to percentages, they are (i) converted to decimal fractions and (ii) multiplied by 100

[1.563]

[13.84]

Determine the dimension marked x in the length of shaft shown in Figure 1.1. The dimensions are in millimetres. [12.52 mm] 82.92 27.41

8.32

x

5 5 = 0.3125, hence corresponds 16 16 to 0.3125 ×100%, i.e. 31.25% 2 (b) Similarly, 1 = 1.4 when expressed as a decimal 5 fraction. 2 Hence 1 = 1.4 × 100% =140% 5 (a)

By division,

Problem 23. It takes 50 minutes to machine a certain part, Using a new type of tool, the time can be reduced by 15%. Calculate the new time taken

34.67

15% of 50 minutes =

750 15 × 50 = 100 100 = 7.5 minutes.

hence the new time taken is 50 − 7.5 = 42.5 minutes.

Figure 1.1

14.

A tank contains 1800 litres of oil. How many tins containing 0.75 litres can be ﬁlled from this tank? [2400]

Alternatively, if the time is reduced by 15%, then it now takes 85% of the original time, i.e. 85% of 85 4250 50 = × 50 = = 42.5 minutes, as above. 100 100 Problem 24. Find 12.5% of £378

1.4

Percentages

Percentages are used to give a common standard and are fractions having the number 100 as their denomi25 1 nators. For example, 25 per cent means i.e. and 100 4 is written 25%. Problem 21. Express as percentages: (a) 1.875 and (b) 0.0125 A decimal fraction is converted to a percentage by multiplying by 100. Thus, (a)

1.875 corresponds to 1.875 ×100%, i.e. 187.5%

(b) 0.0125 corresponds to 0.0125 ×100%, i.e. 1.25% Problem 22. Express as percentages: 5 2 (a) and (b) 1 16 5

12.5% of £378 means ‘per hundred’.

12.5 × 378, since per cent means 100

Hence 12.5% of £378 = 378 = £47.25 8

12.51 1 × 378 = × 378 = 1008 8

Problem 25. Express 25 minutes as a percentage of 2 hours, correct to the nearest 1% Working in minute units, 2 hours =120 minutes. 25 ths of 2 hours. By cancelling, Hence 25 minutes is 120 25 5 = 120 24 5 as a decimal fraction gives 0.2083˙ Expressing 24

9

Section 1

Revision of fractions, decimals and percentages

Section 1

10 Engineering Mathematics Multiplying by 100 to convert the decimal fraction to a percentage gives: 0.2083˙ × 100 = 20.83% Thus 25 minutes is 21% of 2 hours, correct to the nearest 1%. Problem 26. A German silver alloy consists of 60% copper, 25% zinc and 15% nickel. Determine the masses of the copper, zinc and nickel in a 3.74 kilogram block of the alloy By direct proportion: 100% corresponds to 3.74 kg 3.74 = 0.0374 kg 100 60% corresponds to 60 × 0.0374 = 2.244 kg 1% corresponds to

25% corresponds to 25 × 0.0374 = 0.935 kg 15% corresponds to 15 × 0.0374 = 0.561 kg Thus, the masses of the copper, zinc and nickel are 2.244 kg, 0.935 kg and 0.561 kg, respectively. (Check: 2.244 +0.935 + 0.561 =3.74) Now try the following exercise Exercise 4

Further problems percentages

1.

Convert to percentages: (a) 0.057 (b) 0.374 (c) 1.285 [(a) 5.7% (b) 37.4% (c) 128.5%]

2.

Express as percentages, correct to 3 signiﬁcant ﬁgures: (a)

19 11 7 (b) (c) 1 33 24 16 [(a) 21.2% (b) 79.2% (c) 169%]

3.

Calculate correct to 4 signiﬁcant ﬁgures: (a) 18% of 2758 tonnes (b) 47% of 18.42 grams (c) 147% of 14.1 seconds [(a) 496.4 t (b) 8.657 g (c) 20.73 s]

4. When 1600 bolts are manufactured, 36 are unsatisfactory. Determine the percentage unsatisfactory. [2.25%] 5. Express: (a) 140 kg as a percentage of 1 t (b) 47 s as a percentage of 5 min (c) 13.4 cm as a percentage of 2.5 m [(a) 14% (b) 15.67% (c) 5.36%] 6. A block of monel alloy consists of 70% nickel and 30% copper. If it contains 88.2 g of nickel, determine the mass of copper in the block. [37.8 g] 7. A drilling machine should be set to 250 rev/min. The nearest speed available on the machine is 268 rev/min. Calculate the percentage over speed. [7.2%] 8. Two kilograms of a compound contains 30% of element A, 45% of element B and 25% of element C. Determine the masses of the three elements present. [A 0.6 kg, B 0.9 kg, C 0.5 kg] 9. A concrete mixture contains seven parts by volume of ballast, four parts by volume of sand and two parts by volume of cement. Determine the percentage of each of these three constituents correct to the nearest 1% and the mass of cement in a two tonne dry mix, correct to 1 signiﬁcant ﬁgure. [54%, 31%, 15%, 0.3 t] 10. In a sample of iron ore, 18% is iron. How much ore is needed to produce 3600 kg of iron? [20 000 kg] 11. A screws’ dimension is 12.5 ±8% mm. Calculate the possible maximum and minimum length of the screw. [13.5 mm, 11.5 mm] 12. The output power of an engine is 450 kW. If the efﬁciency of the engine is 75%, determine the power input. [600 kW]

Chapter 2

Indices, standard form and engineering notation 2.1

Indices

The lowest factors of 2000 are 2 × 2 × 2 ×2 × 5 × 5 ×5. These factors are written as 24 × 53, where 2 and 5 are called bases and the numbers 4 and 5 are called indices. When an index is an integer it is called a power. Thus, 24 is called ‘two to the power of four’, and has a base of 2 and an index of 4. Similarly, 53 is called ‘ﬁve to the power of 3’ and has a base of 5 and an index of 3. Special names may be used when the indices are 2 and 3, these being called ‘squared’ and ‘cubed’, respectively. Thus 72 is called ‘seven squared’ and 93 is called ‘nine cubed’. When no index is shown, the power is 1, i.e. 2 means 21 .

Reciprocal The reciprocal of a number is when the index is −1 and its value is given by 1, divided by the base. Thus the reciprocal of 2 is 2−1 and its value is 12 or 0.5. Similarly, the reciprocal of 5 is 5−1 which means 15 or 0.2

Square root The square root of a number is when the index is 12 , √ and the square root of 2 is written as 21/2 or 2. The value of a square root is the value of the base which when multiplied √ by itself gives the number. Since 3√×3 = 9, then 9 = 3. However, (−3) × (−3) = 9, so 9 = −3. There are always two answers when ﬁnding the square root of a number and this is shown by putting both a + and a − sign in√front of the answer to√a square root problem. Thus 9 = ±3 and 41/2 = 4 = ±2, and so on. DOI: 10.1016/B978-0-08-096562-8.00002-X

Laws of indices When simplifying calculations involving indices, certain basic rules or laws can be applied, called the laws of indices. These are given below. (i) When multiplying two or more numbers having the same base, the indices are added. Thus 32 × 34 = 32+4 = 36 (ii) When a number is divided by a number having the same base, the indices are subtracted. Thus 35 = 35−2 = 33 32 (iii) When a number which is raised to a power is raised to a further power, the indices are multiplied. Thus (35 )2 = 35×2 = 310 (iv) When a number has an index of 0, its value is 1. Thus 30 = 1 (v) A number raised to a negative power is the reciprocal of that number raised to a positive power. 1 = 23 Thus 3−4 = 314 Similarly, 2−3 (vi) When a number is raised to a fractional power the denominator of the fraction is the root of the number and the numerator is the power. √ 3 Thus 82/3 = 82 = (2)2 = 4 √ √ 2 and 251/2 = 251 = 251 = ±5 √ √ ≡2 ) (Note that

Section 1

12 Engineering Mathematics 2.2

Worked problems on indices

Problem 1. Evaluate: (a) 52 × 53 , (b) 32 × 34 × 3 and (c) 2 × 22 × 25

From the laws of indices: (102 )3 10(2×3) 106 = = 104 × 102 10(4+2) 106 = 106−6 = 100 = 1 Problem 6.

Find the value of: 2 3 × 24 (32 )3 (a) 7 and (b) 5 2 ×2 3 × 39

From law (i): (a) 52 × 53 = 5(2+3) = 55 = 5 × 5× 5× 5 × 5 = 3125 (b) 32 × 34 × 3 = 3(2+4+1) = 37 = 3 ×3 × · · · to 7 terms = 2187

From the laws of indices: 23 × 24 2(3+4) 27 = = = 27−12 = 2−5 27 × 25 2(7+5) 212 1 1 = 5= 2 32 (32 )3 32×3 36 6−10 −4 = = =3 =3 3 × 39 31+9 310 1 1 = 4= 3 81

(a)

(c) 2 × 22 × 25 = 2(1+2+5) = 28 = 256 Problem 2.

Find the value of: 57 75 (a) 3 and (b) 4 7 5

(b)

From law (ii): (a)

75 = 7(5−3) = 72 = 49 73

(b)

57 = 5(7−4) = 53 = 125 54

Now try the following exercise Exercise 5

Problem 3. Evaluate: (a) 52 × 53 ÷ 54 and (b) (3 × 35 ) ÷ (32 × 33 )

Further problems on indices

In Problems 1 to 10, simplify the expressions given, expressing the answers in index form and with positive indices: 1.

(a) 33 × 34 (b) 42 × 43 ×44

From laws (i) and (ii): (a) 52 × 53 ÷ 54 = =

[(a) 37 (b) 49 ]

52 × 5 3 54 55 54

=

5(2+3) 54

2.

(a) 23 × 2 × 22 (b) 72 × 74 × 7 ×73 [(a) 26 (b) 710 ]

= 5(5−4) = 51 = 5 3 × 35 3(1+5) = 32 × 33 3(2+3) 36 = 5 = 3(6−5) = 31 = 3 3

(b) (3 × 35 ) ÷ (32 × 33 ) =

Problem 4. Simplify: (a) (23 )4 and (b) (32 )5 , expressing the answers in index form.

(a)

4.

(a) 56 ÷ 53 (b) 713/710

5.

(a) (72 )3 (b) (33)2

6.

(a)

From law (iii):

[(a) 2 (b) 35 ] [(a) 53 (b) 73 ] [(a) 76 (b) 36 ]

2 2 × 23 37 × 34 (b) 4 2 35 [(a) 2 (b) 36 ]

(a) (23 )4 = 23×4 = 212 (b) (32 )5 = 32×5 = 310 7. Problem 5.

37 24 (b) 2 3 2 3

3.

Evaluate:

(102 )3 104 × 102

(a)

135 57 (b) 52 × 53 13 ×132 [(a) 52 (b) 132 ]

8.

(a)

(a)

32 × 3−4 5−2 (b) 5−4 33

34

(a)

(d)

9−1/2 =

(b) 1]

(a) 52 (b)

10.

(c)

(b) [(a)

9.

√ 41/2 = 4 = ±2 √ 4 163/4 = 163 = (±2)3 = ±8 (Note that it does not matter whether the 4th root of 16 is found ﬁrst or whether 16 cubed is found ﬁrst — the same answer will result). √ 3 272/3 = 272 = (3)2 = 9

(a)

(9 × 32 )3 (16 × 4)2 (b) 2 (3 × 27) (2 × 8)3

1 35

72 × 7−3 23 × 2−4 × 25 (b) −4 7×7 2 × 2−2 × 26 1 (a) 72 (b) 2

2.3 Further worked problems on indices

1 1 1 1 =± =√ = 91/2 ±3 3 9

Problem 10. Evaluate:

33 × 5 7 53 × 3 4

The laws of indices only apply to terms having the same base. Grouping terms having the same base, and then applying the laws of indices to each of the groups independently gives: 33 57 3 3 × 57 = 4 = 3 = 3(3−4) × 5(7−3) 3 4 5 ×3 3 5 54 625 1 = 3−1 × 54 = 1 = = 208 3 3 3 Problem 8. Find the value of: 23 × 35 × (72 )2 74 × 2 4 × 3 3

and

41.5 × 81/3 8 × 2 16 = 16 = = 2 −2/5 1 2 × 32 4 × 14

Alternatively, 41.5 × 81/3 [(2)2 ]3/2 × (23 )1/3 23 × 2 1 = = 22 × 32−2/5 22 × (25 )−2/5 22 × 2−2 = 23+1−2−(−2) = 24 = 16

Problem 11. Evaluate:

= 2−1 × 32 × 70 1 9 1 = × 32 × 1 = = 4 2 2 2

3 2 × 55 + 33 × 53 34 × 5 4

Dividing each term by the HCF (i.e. highest common factor) of the three terms, i.e. 32 × 53 , gives: 3 2 × 55 + 3 3 × 53 34 × 54

3 2 × 55 33 × 53 + 32 × 53 32 × 53 = 34 × 54 32 × 53 =

23 × 35 × (72 )2 = 23−4 × 35−3 × 72×2−4 7 4 × 24 × 33

41.5 × 81/3 22 × 32−2/5

√ 41.5 = 4√3/2 = 43 = 23 = 8 81/3 = 3 8 = 2, 22 = 4 1 1 1 1 = 2= 32−2/5 = 2/5 = √ 5 2 32 2 4 32

Hence Problem 7. Evaluate:

3(2−2) × 5(5−3) + 3(3−2) × 50 3(4−2) × 5(4−3)

3 0 × 52 + 3 1 × 50 32 × 51 1 × 25 + 3 × 1 28 = = 9×5 45 =

Problem 12. Find the value of: Problem 9. Evaluate: (a)

41/2

(b)

163/4

(c)

272/3

(d)

9−1/2

13

32 × 55 34 × 5 4 + 3 3 × 5 3

Section 1

Indices, standard form and engineering notation

Section 1

14 Engineering Mathematics To simplify the arithmetic, each term is divided by the HCF of all the terms, i.e. 32 × 53 . Thus

Exercise 6

32 × 5 5 34 × 5 4 + 3 3 × 53

=

1. (a)

3(2−2) × 5(5−3) 3(4−2) × 5(4−3) + 3(3−2) × 5(3−3) 3 0 × 52 3 2 × 51 + 3 1 × 50

Further problems on indices

In Problems 1 and 2, simplify the expressions given, expressing the answers in index form and with positive indices:

3 2 × 55 2 3 = 4 34 × 5 3 3 ×5 3 × 53 + 3 2 × 53 3 2 × 53 =

Now try the following exercise

=

25 25 = 45 + 3 48

3 −2 3 4 × 3 5 Problem 13. Simplify: −3 2 5 giving the answer with positive indices A fraction raised to a power means that both the numerator and the denominator of the fraction are raised to 3 4 43 that power, i.e. = 3 3 3 A fraction raised to a negative power has the same value as the inverse of the fraction raised to a positive power. −2 3 1 1 5 2 52 Thus, = 2 = 2 = 1 × 2 = 2 5 3 3 3 3 2 5 5 −3 3 2 5 53 Similarly, = = 3 5 2 2 3 −2 3 4 4 3 52 × × 3 5 33 32 Thus, = −3 53 2 23 5 =

4 3 52 2 3 × × 33 32 53

(22 )3 × 23 = (3+2) 3 × 5(3−2) =

2. (a)

42 × 93 8−2 × 52 × 3−4 (b) 83 × 3 4 252 × 24 × 9−2 1 32 (a) 5 (b) 10 2 × 52 2

1 32

−1

(b) 810.25 1/2 4 (c) 16(−1/4) (d) 9 1 2 (a) 9 (b) 3 (c) (d) 2 3

3. Evaluate (a)

In Problems 4 to 8, evaluate the expressions given.

4.

9 2 × 74 3 4 × 74 + 33 × 7 2

5.

(24 )2 − 3−2 × 44 23 × 162

6.

7.

29 35 × 5

7−2 × 3−2 3 3 × 52 (b) 54 × 3 4 35 × 74 × 7−3 1 1 (a) (b) 3 3 ×52 7 × 37

8.

3 −2 2 1 − 2 3 2 3 5 4 4 3 2 2 9 (32 )3/2 × (81/3 )2 (3)2 × (43 )1/2 × (9)−1/2

147 148

1 9

65 −5 72

[64]

1 4 2

2.4

2.5 Worked problems on standard form

Standard form

A number written with one digit to the left of the decimal point and multiplied by 10 raised to some power is said to be written in standard form. Thus: 5837 is written as 5.837 ×103 in standard form, and 0.0415 is written as 4.15 ×10−2 in standard form. When a number is written in standard form, the ﬁrst factor is called the mantissa and the second factor is called the exponent. Thus the number 5.8 × 103 has a mantissa of 5.8 and an exponent of 103. (i) Numbers having the same exponent can be added or subtracted in standard form by adding or subtracting the mantissae and keeping the exponent the same. Thus: 2.3 × 104 + 3.7 × 104 = (2.3 + 3.7) × 10 = 6.0 × 10 4

and 5.9 × 10

−2

− 4.6 × 10

15

4

Problem 14. Express in standard form: (a) 38.71 (b) 3746 (c) 0.0124 For a number to be in standard form, it is expressed with only one digit to the left of the decimal point. Thus: (a) 38.71 must be divided by 10 to achieve one digit to the left of the decimal point and it must also be multiplied by 10 to maintain the equality, i.e. 38.71 = (b) 3746 = form

−2

= (5.9 − 4.6) × 10−2 = 1.3 × 10−2 When the numbers have different exponents, one way of adding or subtracting the numbers is to express one of the numbers in non-standard form, so that both numbers have the same exponent. Thus: 2.3 × 104 + 3.7 × 103 = 2.3 × 104 + 0.37 × 104 = (2.3 + 0.37) × 104 = 2.67 × 104 Alternatively, 2.3 × 104 + 3.7 × 103 = 23 000 + 3700 = 26 700 = 2.67 × 104 (ii) The laws of indices are used when multiplying or dividing numbers given in standard form. For example, (2.5 × 103 ) × (5 × 102 ) = (2.5 × 5) × (103+2 ) = 12.5 × 105 or 1.25 × 106 Similarly, 6 × 104 6 = × (104−2) = 4 × 102 1.5 1.5 × 102

38.71 ×10 = 3.871 ×10 in standard form 10

3746 × 1000 =3.746 ×103 in standard 1000

(c) 0.0124 =0.0124 ×

100 1.24 = 100 100

= 1.24 ×10−2 in standard form Problem 15. Express the following numbers, which are in standard form, as decimal numbers: (a) 1.725 × 10−2 (b) 5.491 × 104 (c) 9.84 × 100 (a) 1.725 × 10−2 =

1.725 = 0.01725 100

(b) 5.491 × 104 = 5.491 ×10 000 = 54 910 (c) 9.84 × 100 = 9.84 × 1 =9.84 (since 100 = 1) Problem 16. Express in standard form, correct to 3 signiﬁcant ﬁgures: 3 2 9 (a) (b) 19 (c) 741 8 3 16 (a)

3 = 0.375, and expressing it in standard form 8 gives: 0.375 =3.75 × 10−1

2 (b) 19 = 19.6˙ = 1.97 ×10 in standard form, correct 3 to 3 signiﬁcant ﬁgures 9 = 741.5625 =7.42 × 102 in standard form, 16 correct to 3 signiﬁcant ﬁgures

(c) 741

Section 1

Indices, standard form and engineering notation

Section 1

16 Engineering Mathematics Problem 17. Express the following numbers, given in standard form, as fractions or mixed numbers: (a) 2.5 ×10−1 (b) 6.25 × 10−2 (c) 1.354 ×102

2.6 Further worked problems on standard form Problem 18.

(a) 2.5 × 10−1 =

1 2.5 25 = = 10 100 4

(b) 6.25 × 10−2 =

625 1 6.25 = = 100 10 000 16

(c) 1.354 × 102 = 135.4 = 135

2 4 = 135 10 5

Now try the following exercise Exercise 7

Further problems on standard form

In Problems 1 to 4, express in standard form: 1. (a) 73.9 (b) 28.4 (c) 197.72 (a) 7.39 × 10 (b) 2.84 × 10 (c) 1.9772 × 102 2. (a) 2748 (b) 33 170 (c) 274 218 (a) 2.748 × 103 (b) 3.317 × 104 (c) 2.74218 × 105

(a)

(b) 8.3 × 103 + 5.415 ×103 and (c) 9.293 × 102 + 1.3 ×103 expressing the answers in standard form. Numbers having the same exponent can be added or subtracted by adding or subtracting the mantissae and keeping the exponent the same. Thus: (a) 7.9 × 10−2 − 5.4 ×10−2 = (7.9 − 5.4) × 10−2 = 2.5 × 10−2 (b) 8.3 × 103 + 5.415 ×103 = (8.3 + 5.415) ×103 = 13.715 ×103 = 1.3715× 104 in standard form (c) Since only numbers having the same exponents can be added by straight addition of the mantissae, the numbers are converted to this form before adding. Thus: 9.293 × 102 + 1.3 × 103 = 9.293 × 102 + 13 × 102 = (9.293 + 13) × 102

3. (a) 0.2401 (b) 0.0174 (c) 0.00923 (a) 2.401 × 10−1 (b) 1.74 × 10−2 (c) 9.23 × 10−3 4. (a)

1 7 3 1 (b) 11 (c) 130 (d) 2 8 5 32 (a) 5 × 10−1 (b) 1.1875 ×10 (c) 1.306 ×102 (d) 3.125 ×10−2

= 22.293 × 102 = 2.2293 × 103 in standard form. Alternatively, the numbers can be expressed as decimal fractions, giving: 9.293 × 102 + 1.3 × 103 = 929.3 + 1300 = 2229.3

In Problems 5 and 6, express the numbers given as integers or decimal fractions: 5. (a) 1.01 × 103 (b) 9.327 ×102 (c) 5.41 ×

104

(d)

7 × 100

[(a) 1010 (b) 932.7 (c) 54 100 (d) 7] 6. (a) (c)

3.89 ×10−2

(b)

6.741 × 10−1

8 × 10−3 [(a) 0.0389 (b) 0.6741 (c) 0.008]

Find the value of:

7.9 × 10−2 − 5.4 ×10−2

= 2.2293× 103 in standard form as obtained previously. This method is often the ‘safest’ way of doing this type of problem. Problem 19.

Evaluate (a) (3.75 × 103 )(6 × 104)

3.5 × 105 7 × 102 expressing answers in standard form and (b)

(a)

(3.75 × 103 )(6 × 104 ) = (3.75 × 6)(103+4) = 22.50 ×107 = 2.25 × 108

(b)

3.5 × 105 3.5 × 105−2 = 7 7 × 102

2.7 Engineering notation and common preﬁxes Engineering notation is similar to scientiﬁc notation except that the power of ten is always a multiple of 3. For example, 0.00035 = 3.5 × 10−4 in scientiﬁc notation,

= 0.5 × 103 = 5 ×102

but 0.00035 = 0.35 × 10−3 or 350 ×10−6 in engineering notation.

Now try the following exercise Exercise 8

Further problems on standard form

In Problems 1 to 4, ﬁnd values of the expressions given, stating the answers in standard form: 1.

2.

(a) 3.7 × 102 + 9.81 × 102 (b) 1.431 × 10−1 + 7.3 × 10−1 [(a) 1.351 × 103 (b) 8.731 × 10−1 ]

3.

1.7231 × 103

(b)

5.

(a)

6 × 10−3 −5

3 × 10

(b)

5 × 1 000 000 = 5 × 106

5 MV means

= 5 000 000 volts = 3600 ohms 7.5 or 106 7.5 × 10−6 = 0.0000075 coulombs

7.5 μC means (b) 1.1 × 105 ]

(2.4 × 103 )(3 × 10−2 ) (4.8 × 104 )

and

4 mA means

[(a) 2 × 102 (b) 1.5 × 10−3 ]

Write the following statements in standard form: (a) The density of aluminium is 2710 kg m−3 [2.71 × 103 kg m−3 ] (b) Poisson’s ratio for gold is 0.44 [4.4 × 10−1 ] (c) The impedance of free space is 376.73 [3.7673 ×102 ] (d) The electron rest energy is 0.511 MeV [5.11 ×10−1 MeV] (e) Proton charge-mass ratio is 9 5 789 700 C kg−1 [9.57897 ×107 C kg−1 ] (f) The normal volume of a perfect gas is 0.02241 m3 mol−1 [2.241 ×10−2 m3 mol−1 ]

3.6 × 1000 = 3.6 × 103

3.6 k means

3.129 × 10−3 ]

(a) (4.5 ×10−2 )(3 ×103 ) (b) 2 × (5.5 ×104 ) [(a) 1.35 ×102

4.

Units used in engineering and science may be made larger or smaller by using preﬁxes that denote multiplication or division by a particular amount. The eight most common multiples, with their meaning, are listed in Table 2.1, where it is noticed that the preﬁxes involve powers of ten which are all multiples of 3. For example,

(a) 4.831 × 102 + 1.24 × 103 (b) 3.24 × 10−3 − 1.11 × 10−4 [(a)

17

7.5 ÷ 1 000 000 =

4 × 10−3 or = =

4 103

4 = 0.004 amperes 1000

Similarly, 0.00006 J = 0.06 mJ or 60 μJ 5 620 000 N = 5620 kN or 5.62 MN 47 ×104 = 470 000 = 470 k or 0.47 M and

12 × 10−5 A = 0.00012 A = 0.12 mA or 120 μA

A calculator is needed for many engineering calculations, and having a calculator which has an ‘EXP’ and ‘ENG’ function is most helpful. For example, to calculate: 3 × 104 × 0.5 × 10−6 volts, input your calculator in the following order: (a) Enter ‘3’ (b) Press ‘EXP’ (or ×10x ) (c) Enter ‘4’ (d) Press ‘×’ (e) Enter ‘0.5’ (f) Press ‘EXP’ (or ×10x ) (g) Enter ‘−6’ (h)Press ‘=’ 7 Now press the ‘ENG’ The answer is 0.015 V or 200 button, and the answer changes to 15 × 10−3 V.

Section 1

Indices, standard form and engineering notation

18 Engineering Mathematics

Section 1

Table 2.1 Preﬁx

Name

Meaning

T

tera

multiply by 1 000 000 000 000

(i.e. × 1012 )

G

giga

multiply by 1 000 000 000

(i.e. × 109 )

M

mega

multiply by 1 000 000

(i.e. × 106 )

k

kilo

multiply by 1000

(i.e. × 103 )

m

milli

divide by 1000

(i.e. × 10−3 )

μ

micro

divide by 1 000 000

(i.e. × 10−6 )

n

nano

divide by 1 000 000 000

(i.e. × 10−9 )

p

pico

divide by 1 000 000 000 000

(i.e. × 10−12 )

The ‘ENG’ or ‘Engineering’ button ensures that the value is stated to a power of 10 that is a multiple of 3, enabling you, in this example, to express the answer as 15 mV. Now try the following exercise Exercise 9

Further problems on engineering notation and common preﬁxes

1. Express the following in engineering notation and in preﬁx form: (a) (c) (e) (g)

100 000 W (b) 0.00054 A (d) 225 ×10−4 V 15 ×105 35 000 000 000 Hz (f) 1.5 × 10−11 F 0.000017 A (h) 46200 [(a) 100 kW (b) 0.54 mA or 540 μA (c) 1.5 M (d) 22.5 mV (e) 35 GHz (f ) 15 pF (g) 17 μA (h) 46.2 k]

2. Rewrite the following as indicated: (a) 0.025 mA = …….μA (b) 1000 pF =…..nF (c) 62 × 104 V = …….kV (d) 1 250 000 = …..M [(a) 25 μA (b) 1 nF (c) 620 kV (d) 1.25 M] 3. Use a calculator to evaluate the following in engineering notation: (a) 4.5 × 10−7 × 3 × 104 (b)

(1.6 × 10−5 )(25 × 103 ) (100 ×106 ) [(a) 13.5 × 10−3 (b) 4 × 103 ]

Chapter 3

Binary, octal and hexadecimal 3.1

Introduction

All data in modern computers is stored as series of bits, a bit being a binary digit, and can have one of two values, the numbers 0 and 1. The most basic form of representing computer data is to represent a piece of data as a string of 1’s and 0’s, one for each bit. This is called a binary or base-2 number. Because binary notation requires so many bits to represent relatively small numbers, two further compact notations are often used, called octal and hexadecimal. Computer programmers who design sequences of number codes instructing a computer what to do, would have a very difﬁcult task if they were forced to work with nothing but long strings of 1s and 0s, the ‘native language’ of any digital circuit. Octal notation represents data as base-8 numbers with each digit in an octal number representing three bits. Similarly, hexadecimal notation uses base-16 numbers, representing four bits with each digit. Octal numbers use only the digits 0–7, while hexadecimal numbers use all ten base-10 digits (0–9) and the letters A–F (representing the numbers 10–15). This chapter explains how to convert between the decimal, binary, octal and hexadecimal systems.

The binary system of numbers has a radix of 2 and uses only the digits 0 and 1. (a) Conversion of binary to decimal: The decimal number 234.5 is equivalent to 2 × 102 + 3 × 101 + 4 × 100 + 5 × 10−1 i.e. is the sum of term comprising: (a digit) multiplied by (the base raised to some power). In the binary system of numbers, the base is 2, so 1101.1 is equivalent to: 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 + 1 × 2−1 Thus the decimal number equivalent to the binary number 1101.1 is 1 8 + 4 + 0 + 1 + , that is 13.5 2 i.e. 1101.12 = 13.510 , the sufﬁxes 2 and 10 denoting binary and decimal systems of number respectively. Problem 1. Convert 110112 to a decimal number From above: 110112 = 1 × 24 + 1 × 23 + 0 × 22

3.2

Binary numbers

The system of numbers in everyday use is the denary or decimal system of numbers, using the digits 0 to 9. It has ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and is said to have a radix or base of 10. DOI: 10.1016/B978-0-08-096562-8.00004-3

+ 1 × 21 + 1 × 20 = 16 + 8 + 0 + 2 + 1 = 2710

Section 1

20 Engineering Mathematics Problem 2.

Convert 0.10112 to a decimal fraction

0.10112 = 1 × 2−1 + 0 × 2−2 + 1 × 2−3 + 1 × 2−4 1 1 1 = 1× +0× 2 +1× 3 2 2 2 1 + 1× 4 2 1 1 1 = + + 2 8 16 = 0.5 + 0.125 + 0.0625 = 0.687510 Problem 3. number

Convert 101.01012 to a decimal

101.01012 = 1 × 22 + 0 × 21 + 1 × 20 + 0 × 2−1 + 1 × 2−2 + 0 × 2−3 + 1 × 2−4 = 4 + 0 + 1 + 0 + 0.25 + 0 + 0.0625 = 5.312510 Now try the following exercise

5. (a) 11010.11 (b) 10111.011 (c) 110101.0111 (d) 11010101.10111 (a) 26.7510 (b) 23.37510 (c) 53.437510 (d) 213.7187510

(b) Conversion of decimal to binary: An integer decimal number can be converted to a corresponding binary number by repeatedly dividing by 2 and noting the remainder at each stage, as shown below for 3910 2 2 2 2 2 2

39 19 9 4 2 1 0

Remainder 1 1 1 0 0 1

(most 1 0 0 1 1 1 significant bit)

(least significant bit)

The result is obtained by writing the top digit of the remainder as the least signiﬁcant bit, (a bit is a binary digit and the least signiﬁcant bit is the one on the right). The bottom bit of the remainder is the most signiﬁcant bit, i.e. the bit on the left. Thus 3910 = 1001112

Exercise 10

Further problems on conversion of binary to decimal numbers

In Problems 1 to 5, convert the binary numbers given to decimal numbers.

The fractional part of a decimal number can be converted to a binary number by repeatedly multiplying by 2, as shown below for the fraction 0.625 0.625 3 2 5

1. 250

1. (a) 110 (b) 1011 (c) 1110 (d) 1001 [(a) 610 (b) 1110 (c) 1410 (d) 910 ]

0.250 3 2 5

0. 500

2. (a) 10101 (b) 11001 (c) 101101 (d) 110011 [(a) 2110 (b) 2510 (c) 4510 (d) 5110 ]

0.500 3 2 5

1. 000

(most significant bit) .1

3. (a) 101010 (b) 111000 (c) 1000001 (d) 10111000 [(a) 4210 (b) 5610 (c) 6510 (d) 18410] 4. (a) 0.1101 (b) 0.11001 (c) 0.00111 (d) 0.01011 (a) 0.812510 (b) 0.7812510 (c) 0.2187510 (d) 0.3437510

1 (least significant bit)

For fractions, the most signiﬁcant bit of the result is the top bit obtained from the integer part of multiplication by 2. The least signiﬁcant bit of the result is the bottom bit obtained from the integer part of multiplication by 2. Thus 0.62510 = 0.1012

Problem 4. Convert 4710 to a binary number

The fractional part is repeatedly multiplied by 2 giving:

From above, repeatedly dividing by 2 and noting the remainder gives:

0.3125 3 2 5 0.625 3 2 5 0.25 3 2 5 0.5 325

2 47 Remainder 2 23 1 2 11 1 2

5 1

2

2 1

2

1 0

0.625 1.25 0.5 1.0 .0 1 0 1

Thus 58.312510 = 111010.01012

0 1 1

1

1

1

1

Thus 4710 = 1011112

Now try the following exercise Exercise 11

Problem 5. Convert 0.4062510 to a binary number From above, repeatedly multiplying by 2 gives: 0.40625 3 2 5

0. 8125

0.8125

325

1. 625

0.625

325

1. 25

0.25

325

0. 5

0.5

325

1. 0 .0

1

1

1

1.

58 29 14 7 3 1 0

Remainder 0 1 0 1 1 1 1

1

(a) 5

(b) 15

(c) 19 (d) 29 (a) 1012 (c) 100112

1

(a) 31

(b) 42 (c) 57 (d) 63 (b) 1010102 (a) 111112 (c) 1110012 (d) 1111112

3.

(a) 47 (b) 60 (c) 73 (d) 84 (b) 1111002 (a) 1011112 (c) 10010012 (d) 10101002

4.

(a) 0.25 (b) 0.21875 (c) 0.28125 (d) 0.59375 (b) 0.001112 (a) 0.012 (c) 0.010012 (d) 0.100112

5.

(a) 47.40625 (b) 30.8125 (c) 53.90625 (d) 61.65625 (a) 101111.011012 (c) 110101.111012

1

(b) 11112 (d) 111012

2.

The integer part is repeatedly divided by 2, giving: 2 2 2 2 2 2

Further problems on conversion of decimal to binary numbers

In Problems 1 to 5, convert the decimal numbers given to binary numbers.

i.e. 04062510 = 0.011012 Problem 6. Convert 58.312510 to a binary number

21

(b) 11110.11012 (d) 111101.101012

Section 1

Binary, octal and hexadecimal

Section 1

22 Engineering Mathematics (c) Binary addition:

Now try the following exercise

Binary addition of two/three bits is achieved according to the following rules: sum carry 0+0 = 0 0 0+1 = 1 0 1+0 = 1 0 1+1 = 0 1

sum carry 0+0+0 = 0 0 0+0+1 = 1 0 0+1+0 = 1 0 0+1+1 = 0 1 1+0+0 = 1 0 1+0+1 = 0 1 1+1+0 = 0 1 1+1+1 = 1 1

These rules are demonstrated in the following worked problems.

Problem 7.

Perform the binary addition: 1001 + 10110 1001 +10110 11111

Problem 8.

Perform the binary addition: 11111 + 10101 11111 +10101 sum 110100 carry 11111

Problem 9.

Perform the binary addition: 1101001 + 1110101 1101001 +1110101 sum 11011110 carry 1 1 1

Problem 10. Perform the binary addition: 1011101 + 1100001 + 110101 1011101 1100001 +110101 sum 11110011 carry 11111 1

Exercise 12

Further problems on binary addition

Perform the following binary additions: 1.

10 + 11

2.

101 + 110

[1011]

3.

1101 + 111

[10100]

4.

1111 + 11101

5.

110111 + 10001

6.

10000101 + 10000101

[100001010]

7.

11101100 + 111001011

[1010110111]

8.

110011010 + 11100011

[1001111101]

9.

10110 + 1011 + 11011

[101]

[101100] [1001000]

[111100]

10. 111 + 10101 + 11011

[110111]

11. 1101 + 1001 + 11101

[110011]

12. 100011 + 11101 + 101110

3.3

[1101110]

Octal numbers

For decimal integers containing several digits, repeatedly dividing by 2 can be a lengthy process. In this case, it is usually easier to convert a decimal number to a binary number via the octal system of numbers. This system has a radix of 8, using the digits 0, 1, 2, 3, 4, 5, 6 and 7. The denary number equivalent to the octal number 43178 is 4 × 8 3 + 3 × 8 2 + 1 × 81 + 7 × 8 0 i.e.

4 × 512 + 3 × 64 + 1 × 8 + 7 × 1 or 225510

An integer decimal number can be converted to a corresponding octal number by repeatedly dividing by 8 and noting the remainder at each stage, as shown below for 49310 8 493 Remainder 8 61 5 8

7 5 0 7 7

Thus 49310 = 7558

5

5

The fractional part of a decimal number can be converted to an octal number by repeatedly multiplying by 8, as shown below for the fraction 0.437510

23

Dividing repeatedly by 8, and noting the remainder gives: 8 3714 Remainder

0.4375 3 8 5

3. 5

385

4. 0

0.5

.3

8 464 2 8

58 0

8

7 2 0 7

4

7

For fractions, the most signiﬁcant bit is the top integer obtained by multiplication of the decimal fraction by 8, thus 0.437510 = 0.348 The natural binary code for digits 0 to 7 is shown in Table 3.1, and an octal number can be converted to a binary number by writing down the three bits corresponding to the octal digit. Thus 4378 = 100 011 1112

From Table 3.1,

2

2

72028 = 111 010 000 0102 371410 = 111 010 000 0102

i.e.

Problem 12. Convert 0.5937510 to a binary number, via octal Multiplying repeatedly by 8, and noting the integer values, gives: 0.59375 3 8 5 0.75 385

and 26.358 = 010 110.011 1012

4.75 6.00 .4 6

Table 3.1 Octal digit

Natural binary number

000

1

001

2

010

3

011

4

100

5

101

6

110

7

111

Thus 0.5937510 = 0.468 From Table 3.1,

The ‘0’ on the extreme left does not signify anything, thus 26.358 = 10 110.011 1012 Conversion of decimal to binary via octal is demonstrated in the following worked problems. Problem 11. Convert 371410 to a binary number, via octal

i.e.

0.468 = 0.100 1102 0.5937510 = 0.100 112

Problem 13. Convert 5613.9062510 to a binary number, via octal The integer part is repeatedly divided by 8, noting the remainder, giving: 8 5613 Remainder 8 701 5 8 87 5 8 10 7 8 1 2 0 1 1

2

7

5

5

This octal number is converted to a binary number, (see Table 3.1) 127558 = 001 010 111 101 1012 i.e.

561310 = 1 010 111 101 1012

Section 1

Binary, octal and hexadecimal

Section 1

24 Engineering Mathematics The fractional part is repeatedly multiplied by 8, and noting the integer part, giving: 0.90625 3 8 5 0.25 385

7.25 2.00 .7 2

This octal fraction is converted to a binary number, (see Table 3.1) 0.728 = 0.111 0102 i.e.

0.9062510 = 0.111 012

Thus, 5613.9062510 = 1 010 111 101 101.111 012 Problem 14. Convert 11 110 011.100 012 to a decimal number via octal Grouping the binary number in three’s from the binary point gives: 011 110 011.100 0102 Using Table 3.1 to convert this binary number to an octal number gives: 363.428 and 363.428 = 3 × 82 + 6 × 81 + 3 × 80 + 4 × 8−1 + 2 × 8−2 = 192 + 48 + 3 + 0.5 + 0.03125

3. (a) 247.09375 (b) 514.4375 (c) 1716.78125 ⎤ ⎡ (a) 11110111.000112 ⎣ (b) 1000000010.01112 ⎦ (c) 11010110100.110012 4. Convert the following binary numbers to decimal numbers via octal: (a) 111.011 1 (b) 101 001.01 (c) 1 110 011 011 010.001 1 (a) 7.437510 (b) 41.2510 (c) 7386.187510

3.4

The hexadecimal system is particularly important in computer programming, since four bits (each consisting of a one or zero) can be succinctly expressed using a single hexadecimal digit. Two hexadecimal digits represent numbers from 0 to 255, a common range used, for example, to specify colours. Thus, in the HTML language of the web, colours are speciﬁed using three pairs of hexadecimal digits RRGGBB, where RR is the amount of red, GG the amount of green, and BB the amount of blue. A hexadecimal numbering system has a radix of 16 and uses the following 16 distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F

= 243.5312510 Now try the following exercise Exercise 13

Further problems on conversion between decimal and binary numbers via octal

Hexadecimal numbers

‘A’ corresponds to 10 in the denary system, B to 11, C to 12, and so on. (a) Converting from hexadecimal to decimal: For example 1A16 = 1 × 161 + A × 160

In Problems 1 to 3, convert the decimal numbers given to binary numbers, via octal. 1. (a) 343 (b) 572 (c) 1265 (b) 10001111002 (a) 1010101112 (c) 100111100012 2. (a) 0.46875 (b) 0.6875 (c) 0.71875 (a) 0.011112 (b) 0.10112 (c) 0.101112

= 1 × 161 + 10 × 1 = 16 + 10 = 26 i.e.

1A16 = 2610

Similarly, 2E16 = 2 × 161 + E × 160 = 2 × 161 + 14 × 160 = 32 + 14 = 4610 and

1BF16 = 1 × 162 + B × 161 + F × 160 = 1 × 162 + 11 × 161 + 15 × 160 = 256 + 176 + 15 = 44710

Table 3.2 compares decimal, binary, octal and hexadecimal numbers and shows, for example, that

Table 3.2 Decimal

2310 = 101112 = 278 = 1716

Binary

Octal

Hexadecimal

0000

1

0001

1

1

2

0010

2

2

3

0011

3

3

4

0100

4

4

5

0101

5

5

6

0110

6

6

7

0111

7

7

8

1000

10

8

9

1001

11

9

10

1010

12

A

11

1011

13

B

12

1100

14

C

13

1101

15

D

14

1110

16

E

15

1111

17

F

16

10000

20

10

Problem 17. Convert 1A4E16 into a denary number

17

10001

21

11

18

10010

22

12

1A4E16

19

10011

23

13

20

10100

24

14

21

10101

25

15

= 1 × 4096 + 10 × 256 + 4 × 16 + 14 × 1

22

10110

26

16

= 4096 + 2560 + 64 + 14 = 6734

23

10111

27

17

24

11000

30

18

25

11001

31

19

26

11010

32

1A

27

11011

33

1B

28

11100

34

1C

29

11101

35

1D

30

11110

36

1E

31

11111

37

1F

32

100000

40

20

Problem 15. Convert the following hexadecimal numbers into their decimal equivalents: (a) 7A16 (b) 3F16 (a)

7A16 = 7 × 161 + A × 160 = 7 ×16 +10 × 1 = 112 +10 =122 Thus 7A16 = 12210

(b) 3F16 = 3 × 161 + F × 160 = 3 × 16 +15 ×1 = 48 +15 =63 Thus, 3F16 = 6310 Problem 16. Convert the following hexadecimal numbers into their decimal equivalents: (a) C916 (b) BD16 (a)

C916 = C × 161 + 9 ×160 = 12 × 16 +9 × 1 = 192 +9 =201 Thus C916 = 20110

(b) BD16 = B × 161 + D × 160 = 11 ×16 +13 × 1 = 176 +13 =189 Thus, BD16 = 18910

= 1 × 163 + A × 162 + 4 × 161 + E × 160 = 1 × 16 + 10 × 16 + 4 × 16 + 14 × 16 3

2

1

Thus, 1A4E16 = 673410 (b) Converting from decimal to hexadecimal: This is achieved by repeatedly dividing by 16 and noting the remainder at each stage, as shown below for 2610. 16 26 Remainder 16 1 10 ; A16 0 1 ; 116 most significant bit

Hence 2610 = 1A16

1 A

least significant bit

25

Section 1

Binary, octal and hexadecimal

Section 1

26 Engineering Mathematics Similarly, for 44710

Now try the following exercise

16 447 Remainder 16

27 15 ; F16

16

1 11 ; B16 0

Exercise 14

Further problems on hexadecimal numbers

In Problems 1 to 4, convert the given hexadecimal numbers into their decimal equivalents.

1 ; 116 1 B F

Thus 44710 = 1BF16 Problem 18. Convert the following decimal numbers into their hexadecimal equivalents: (a) 3710 (b) 10810

1. E716

[23110]

2. 2C16

[4410]

3. 9816

[15210]

4. 2F116

[75310]

In Problems 5 to 8, convert the given decimal numbers into their hexadecimal equivalents. 5. 5410

[3616]

6. 20010

[C816]

7. 9110

[5B16]

8. 23810

[EE16 ]

(a) 16 37 Remainder 16 2 5 5 516

(c) Converting from binary to hexadecimal:

0 2 5 216 2

5

most significant bit

least significant bit

Hence 3710 = 2516

The binary bits are arranged in groups of four, starting from right to left, and a hexadecimal symbol is assigned to each group. For example, the binary number 1110011110101001 is initially grouped in fours as:

(b) 16 108 Remainder 16

1110 0111 1010 1001

and a hexadecimal symbol

6 12 5 C16 0 6 5 616

assigned to each group as E

7

6 C

9

from Table 3.2 Hence 11100111101010012 = E7A916

Hence 10810 = 6C16 Problem 19. Convert the following decimal numbers into their hexadecimal equivalents: (a) 16210 (b) 23910

Problem 20. Convert the following binary numbers into their hexadecimal equivalents: (a) 110101102 (b) 11001112 (a)

(a) 16 162 Remainder 16 10 2 5 216 0 10 5 A16 A 2

Hence 16210 = A216 (b) 16 239 Remainder 16 14 15 5 F16 0 14 5 E16 E F

Hence 23910 = EF16

A

Grouping bits in fours from the right gives: 1101 0110 and assigning hexadecimal symbols to each group gives: D 6 from Table 3.2 Thus, 110101102 = D616

(b) Grouping bits in fours from the right gives: 0110 0111 and assigning hexadecimal symbols to each group gives: 6 7 from Table 3.2 Thus, 11001112 = 6716

Problem 21. Convert the following binary numbers into their hexadecimal equivalents: (a) 110011112 (b) 1100111102 (a)

Problem 23. Convert the following hexadecimal numbers into their binary equivalents: (a) 7B16 (b) 17D16

Grouping bits in fours from the

(a)

right gives:

1100 1111

Spacing out hexadecimal digits gives:

and assigning hexadecimal

7

C F from Table 3.2

binary gives:

Thus, 110011112 = CF16

0111 1011 from Table 3.2

Thus, 7B16 = 11110112

(b) Grouping bits in fours from

(b) Spacing out hexadecimal 0001 1001 1110

digits gives:

and assigning hexadecimal symbols to each group gives:

1

7

1 9 E from Table 3.2

binary gives:

0001 0111 1101 from Table 3.2

Thus, 17D16 = 1011111012

(d) Converting from hexadecimal to binary:

Now try the following exercise

The above procedure is reversed, thus, for example, Exercise 15

6CF316 = 0110 1100 1111 0011 from Table 3.2 i.e. 6CF316 = 1101100111100112

(a) Spacing out hexadecimal digits gives:

3

F

and converting each into 0011 1111 from Table 3.2

Thus, 3F16 = 1111112 (b) Spacing out hexadecimal digits A

6

and converting each into binary gives: Thus, A616 = 101001102

Further problems on hexadecimal numbers

In Problems 1 to 4, convert the given binary numbers into their hexadecimal equivalents.

Problem 22. Convert the following hexadecimal numbers into their binary equivalents: (a) 3F16 (b) A616

gives:

D

and converting each into

Thus, 1100111102 = 19E16

binary gives:

B

and converting each into

symbols to each group gives:

the right gives:

27

Section 1

Binary, octal and hexadecimal

1010 0110 from Table 3.2

1.

110101112

[D716]

2.

111010102

[EA16 ]

3.

100010112

[8B16]

4.

101001012

[A516]

In Problems 5 to 8, convert the given hexadecimal numbers into their binary equivalents. [1101112]

5.

3716

6.

ED16

[111011012]

7.

9F16

[100111112]

8.

A2116

[1010001000012]

Chapter 4

Calculations and evaluation of formulae 4.1

Errors and approximations

(i) In all problems in which the measurement of distance, time, mass or other quantities occurs, an exact answer cannot be given; only an answer which is correct to a stated degree of accuracy can be given. To take account of this an error due to measurement is said to exist. (ii) To take account of measurement errors it is usual to limit answers so that the result given is not more than one signiﬁcant ﬁgure greater than the least accurate number given in the data. (iii) Rounding-off errors can exist with decimal fractions. For example, to state that π = 3.142 is not strictly correct, but ‘π = 3.142 correct to 4 signiﬁcant ﬁgures’ is a true statement. (Actually, π = 3.14159265. . .) (iv) It is possible, through an incorrect procedure, to obtain the wrong answer to a calculation. This type of error is known as a blunder. (v) An order of magnitude error is said to exist if incorrect positioning of the decimal point occurs after a calculation has been completed. (vi) Blunders and order of magnitude errors can be reduced by determining approximate values of calculations. Answers which do not seem feasible must be checked and the calculation must be repeated as necessary. An engineer will often need to make a quick mental approximation for a calculation. For 49.1 × 18.4 × 122.1 may be approxiexample, 61.2 × 38.1 DOI: 10.1016/B978-0-08-096562-8.00004-3

50 × 20 × 120 and then, by cancelling, 60 × 40 1 50 × 1 20 × 120 2 = 50. An accurate answer 1 60 × 40 21 somewhere between 45 and 55 could therefore be expected. Certainly an answer around 500 or 5 would not be expected. Actually, by calculator 49.1 × 18.4 × 122.1 = 47.31, correct to 4 signi61.2 × 38.1 ﬁcant ﬁgures. mated to

Problem 1. The area A of a triangle is given by 1 A = bh. The base b when measured is found to be 2 3.26 cm, and the perpendicular height h is 7.5 cm. Determine the area of the triangle. 1 1 Area of triangle = bh = × 3.26 × 7.5 2 2 = 12.225 cm2 (by calculator). 1 The approximate values is × 3 × 8 = 12 cm 2 , so 2 there are no obvious blunder or magnitude errors. However, it is not usual in a measurement type problem to state the answer to an accuracy greater than 1 signiﬁcant ﬁgure more than the least accurate number in the data: this is 7.5 cm, so the result should not have more than 3 signiﬁcant ﬁgures. Thus, area of triangle = 12.2 cm2 Problem 2. State which type of error has been made in the following statements: (a) 72 × 31.429 = 2262.9 (b) 16 × 0.08 × 7 = 89.6

(c) 11.714 × 0.0088 = 0.3247 correct to 4 decimal places. (d)

(a)

i.e.

2.19 × 203.6 × 17.91 ≈ 75.3, 12.1 × 8.76 correct to 3 signiﬁcant ﬁgures.)

29.74 × 0.0512 = 0.12, correct to 2 11.89 signiﬁcant ﬁgures. 72 × 31.429 = 2262.888 (by calculator), hence a rounding-off error has occurred. The answer should have stated: 72 × 31.429 = 2262.9, correct to 5 signiﬁcant ﬁgures or 2262.9, correct to 1 decimal place.

32 × 7 8 ×7= (b) 16 × 0.08 × 7 = 16 × 100 25

(By

(c)

(d)

Exercise 16

hence no order of magnitude error has occurred. 29.74 × 0.0512 = 0.128 correct to 3 However, 11.89 signiﬁcant ﬁgures, which equals 0.13 correct to 2 signiﬁcant ﬁgures.

1.

25 × 0.06 × 1.4 = 0.21 [Order of magnitude error]

2.

137 × 6.842 = 937.4 ⎡ ⎤ Rounding-off error–should add ‘correct ⎣ to 4 signiﬁcant ﬁgures’ or ‘correct to ⎦ 1 decimal place’

3. 4.

5.

Hence a rounding-off error has occurred. Problem 3. Without using a calculator, determine an approximate value of: 2.19 × 203.6 × 17.91 11.7 × 19.1 (b) (a) 9.3 × 5.7 12.1 × 8.76 (a)

10 × 20 11.7 × 19.1 is approximately equal to 9.3 × 5.7 10 × 5

24 × 0.008 = 10.42 12.6

[Blunder]

For a gas pV = c. When pressure p = 1 03 400 Pa and V = 0.54 m3 then c = 55 836 Pa m3. Measured values, hence c = 55 800 Pa m3 4.6 × 0.07 = 0.225 52.3 × 0.274 ⎡ ⎤ Order of magnitude error and rounding⎢ off error–should be 0.0225, correct to 3 ⎥ ⎢ ⎥ ⎣ signiﬁcant ﬁgures or 0.0225, ⎦ correct to 4 decimal places

In Problems 6 to 8, evaluate the expressions approximately, without using a calculator. 6.

4.7 × 6.3

7.

2.87 × 4.07 6.12 × 0.96

i.e. about 4 11.7 × 19.1 = 4.22, correct to 3 9.3 × 5.7 signiﬁcant ﬁgures.)

Further problems on errors

In Problems 1 to 5 state which type of error, or errors, have been made:

11.714 × 0.0088 is approximately equal to 12 × 9 × 10−3, i.e. about 108 × 10−3 or 0.108. Thus a blunder has been made. 29.74 × 0.0512 30 × 5 × 10−2 ≈ 11.89 12 150 15 1 = = = or 0.125 2 12 × 10 120 8

calculator,

Now try the following exercise

24 224 = 8 = 8.96 = 25 25 Hence an order of magnitude error has occurred.

2.19 × 203.6 × 17.91 ≈ 80 12.1 × 8.76

[≈30 (29.61, by calculator)]

≈2 (1.988, correct to 4 s.f., by calculator)

(By calculator,

(b)

2.19 × 203.6 × 17.91 2 × 20 200 × 202 ≈ 12.1 × 8.76 1 10 × 101 = 2 × 20 × 2 after cancelling,

8.

72.1 × 1.96 × 48.6 139.3 × 5.2 ≈10 (9.481, correct to 4 s.f., by calculator)

Section 1

29

Calculations and evaluation of formulae

Section 1

30 Engineering Mathematics 4.2

Use of calculator

The most modern aid to calculations is the pocket-sized electronic calculator. With one of these, calculations can be quickly and accurately performed, correct to about 9 signiﬁcant ﬁgures. The scientiﬁc type of calculator has made the use of tables and logarithms largely redundant. To help you to become competent at using your calculator check that you agree with the answers to the following problems: Problem 4. Evaluate the following, correct to 4 signiﬁcant ﬁgures: (a) 4.7826 + 0.02713 (b) 17.6941 − 11.8762 (c) 21.93 × 0.012981

(a)

1 = 0.01896453 . . . = 0.019, correct to 3 52.73 decimal places

(b)

1 = 36.3636363 . . . = 36.364, correct to 3 0.0275 decimal places

(c)

1 1 + = 0.71086624 . . . = 0.711, correct 4.92 1.97 to 3 decimal places

Problem 7. Evaluate the following, expressing the answers in standard form, correct to 4 signiﬁcant ﬁgures: (a) (0.00451)2 (b) 631.7 − (6.21 + 2.95)2 2 2 (c) 46.27 − 31.79 (a)

(a)

4.7826 + 0.02713 = 4.80973 = 4.810, correct to 4 signiﬁcant ﬁgures

(b) 17.6941 − 11.8762 = 5.8179 = 5.818, correct to 4 signiﬁcant ﬁgures (c)

21.93 × 0.012981 = 0.2846733. . . = 0.2847, correct to 4 signiﬁcant ﬁgures

Problem 5. Evaluate the following, correct to 4 decimal places: (a) 46.32 × 97.17 × 0.01258 (b) (c) (a)

4.621 23.76

1 (62.49 × 0.0172) 2 46.32 × 97.17 × 0.01258 = 56.6215031. . . = 56.6215, correct to 4 decimal places

(b)

4.621 = 0.19448653. . . = 0.1945, correct to 4 23.76 decimal places

(c)

1 (62.49 × 0.0172) = 0.537414 = 0.5374, 2 correct to 4 decimal places

Problem 6. Evaluate the following, correct to 3 decimal places: (a)

1 1 1 1 (b) (c) + 52.73 0.0275 4.92 1.97

(0.00451)2 = 2.03401 × 10−5 = 2.034 × 10−5 , correct to 4 signiﬁcant ﬁgures

(b) 631.7 − (6.21 + 2.95)2 = 547.7944 = 5.477944 × 102 = 5.478 × 102 , correct to 4 signiﬁcant ﬁgures (c)

46.272 − 31.792 = 1130.3088 = 1.130 × 103 , correct to 4 signiﬁcant ﬁgures

Problem 8. Evaluate the following, correct to 3 decimal places: (2.37)2 5.40 2 3.60 2 (a) + (b) 0.0526 1.92 2.45 15 (c) 7.62 − 4.82 (2.37)2 = 106.785171. . . = 106.785, correct to 3 0.0526 decimal places 2 2 3.60 5.40 (b) + = 8.37360084. . . = 8.374, 1.92 2.45 correct to 3 decimal places 15 = 0.43202764. . . = 0.432, correct to (c) 7.62 − 4.82 3 decimal places (a)

Problem 9. Evaluate the following, correct to 4 signiﬁcant ﬁgures: √ √ √ (a) 5.462 (b) 54.62 (c) 546.2

Calculations and evaluation of formulae

Problem 10. Evaluate the following, correct to 3 decimal places: √ √ √ (b) 52.91 − 31.76 (a) 0.007328 √ (c) 1.6291 × 104 √

0.007328 = 0.08560373 = 0.086, correct to 3 decimal places √ √ 52.91 − 31.76 = 1.63832491 . . . = 1.638, (b) correct to 3 decimal places √ √ (c) 1.6291 × 104 = 16291 = 127.636201. . . = 127.636, correct to 3 decimal places (a)

Problem 11. Evaluate the following, correct to 4 signiﬁcant ﬁgures: √ (a) 4.723 (b) (0.8316)4 (c) 76.212 − 29.102

Problem 13. Evaluate the following, expressing the answers in standard form, correct to 4 decimal places: (a) (5.176 × 10−3)2 4 1.974 × 101 × 8.61 × 10−2 (b) 3.462 √ (c) 1.792 × 10−4 (5.176 × 10−3)2 = 2.679097. . . × 10−5 = 2.6791 × 10−5 , correct to 4 decimal places 4 1.974 × 101 × 8.61 × 10−2 (b) = 0.05808887. . . 3.462 = 5.8089 × 10−2 , correct to 4 decimal places √ 1.792 ×10−4 = 0.0133865. . . = 1.3387×10−2 , (c) correct to 4 decimal places (a)

Now try the following exercise Exercise 17

Further problems on the use of a calculator

4.723 = 105.15404. . . = 105.2, correct to 4 signiﬁcant ﬁgures

In Problems 1 to 9, use a calculator to evaluate the quantities shown correct to 4 signiﬁcant ﬁgures:

(b) (0.8316)4 = 0.47825324. . . = 0.4783, correct to 4 signiﬁcant ﬁgures √ (c) 76.212 − 29.102 = 70.4354605. . . = 70.44, correct to 4 signiﬁcant ﬁgures

1. (a) 3.2492 (b) 73.782 (c) 311.42 (d) 0.06392 (a) 10.56 (b) 5443 (c) 96970 (d) 0.004083 √ √ √ 2. (a) √4.735 (b) 35.46 (c) 73 280 (d) 0.0256 (a) 2.176 (b) 5.955 (c) 270.7 (d) 0.1600

(a)

Problem 12. Evaluate the following, correct to 3 signiﬁcant ﬁgures: √ 6.092 (a) (b) 3 47.291 √ 25.2 × 7 √ (c) 7.2132 + 6.4183 + 3.2914 (a)

(b) (c)

6.092 √ = 0.74583457. . . = 0.746, correct 25.2 × 7 to 3 signiﬁcant ﬁgures

√ 3 47.291 = 3.61625876. . . = 3.62, correct to 3 signiﬁcant ﬁgures √ 7.2132 + 6.4183 + 3.2914 = 20.8252991. . ., = 20.8 correct to 3 signiﬁcant ﬁgures

1 1 1 (b) (c) 7.768 48.46 0.0816 1 (d) 1.118 (a) 0.1287 (b) 0.02064 (c) 12.25 (d) 0.8945

3. (a)

4. (a) 127.8 × 0.0431 × 19.8 (b) 15.76 ÷ 4.329 [(a) 109.1 5. (a)

137.6 552.9

(b)

(b) 3.641]

11.82 × 1.736 0.041 [(a) 0.2489 (b) 500.5]

Section 1

√

5.462 = 2.3370922. . . = 2.337, correct to 4 signiﬁcant ﬁgures √ 54.62 = 7.39053448. . . = 7.391, correct to 4 (b) signiﬁcant ﬁgures √ 546.2 = 23.370922 . . . = 23.37, correct to 4 (c) signiﬁcant ﬁgures (a)

31

Section 1

32 Engineering Mathematics 6.

(a) 13.63 (b) 3.4764 (c) 0.1245 [(a) 2515 (b) 146.0 (c) 0.00002932]

7.

8.

24.68 × 0.0532 7.412 4 0.2681 × 41.22 (b) 32.6 × 11.89 [(a) 0.005559 (b) 1.900]

(a)

(a)

14.323 21.682

9.

3

(a) (b)

(b)

4.8213 2 17.33 − 15.86 × 11.6 [(a) 6.248 (b) 0.9630]

(15.62)2 √ 29.21 × 10.52

√ 6.9212 + 4.8163 − 2.1614 [(a) 1.605 (b) 11.74]

10. Evaluate the following, expressing the answers in standard form, correct to 3 decimal places: (a) (8.291 × 10−2)2 √ (b) 7.623 × 10−3 [(a) 6.874 × 10−3

(b) 8.731 × 10−2]

Calculate: (a) how many French euros £27.80 will buy (b) the number of Japanese yen which can be bought for £23 (c)

the pounds sterling which can be exchanged for 7114.80 Norwegian kronor

(d) the number of American dollars which can be purchased for £90, and (e)

(a)

the pounds sterling which can be exchanged for 2990 Swiss francs £1 = 1.46 euros, hence £27.80 = 27.80 × 1.46 euros = 40.59 euros

(b) £1 = 220 yen, hence £23 = 23 × 220 yen = 5060 yen £1 = 12.10 kronor, hence 7114.80 = £588 7114.80 kronor = £ 12.10 (d) £1 = 1.95 dollars, hence £90 = 90 × 1.95 dollars = $175.50 (c)

(e)

£1 = 2.30 Swiss francs, hence 2990 = £1300 2990 franc = £ 2.30

Problem 15. Some approximate imperial to metric conversions are shown in Table 4.2

4.3

Conversion tables and charts

It is often necessary to make calculations from various conversion tables and charts. Examples include currency exchange rates, imperial to metric unit conversions, train or bus timetables, production schedules and so on. Problem 14. Currency exchange rates for ﬁve countries are shown in Table 4.1 Table 4.1 France

£1 = 1.46 euros

Japan

£1 = 220 yen

Norway

£1 = 12.10 kronor

Switzerland

£1 = 2.30 francs

U.S.A.

£1 = 1.95 dollars ($)

Table 4.2 length

1 inch = 2.54 cm 1 mile = 1.61 km

weight

2.2 lb = 1 kg (1 lb = 16 oz)

capacity

1.76 pints = 1 litre (8 pints = 1 gallon)

Use the table to determine: (a) the number of millimetres in 9.5 inches, (b) a speed of 50 miles per hour in kilometres per hour, (c) the number of miles in 300 km, (d) the number of kilograms in 30 pounds weight, (e) the number of pounds and ounces in 42 kilograms (correct to the nearest ounce), (f ) the number of litres in 15 gallons, and (g) the number of gallons in 40 litres.

(a)

9.5 inches = 9.5 × 2.54 cm = 24.13 cm 24.13 cm = 24.13 × 10 mm = 241.3 mm

(b)

50 m.p.h. = 50 × 1.61 km/h = 80.5 km/h

(c)

300 km =

(d)

30 lb =

(e)

42 kg = 42 × 2.2 lb = 92.4 lb

2.

300 miles = 186.3 miles 1.61

(g)

15 gallons = 15 × 8 pints = 120 pints 120 litres = 68.18 litres 1.76

40 litres = 40 × 1.76 pints = 70.4 pints 70.4 pints =

70.4 gallons = 8.8 gallons 8

3.

Now try the following exercise Exercise 18 1.

Weight

1 kg = 2.2 lb (1 lb = 16 ounces)

Deduce the following information from the train timetable shown in Table 4.3:

(b) A girl leaves Hunts Cross at 8.17 a.m. and travels to Manchester Oxford Road. How long does the journey take? What is the average speed of the journey?

Currency exchange rates listed in a newspaper included the following: £1 = 1.48 euro £1 = 225 yen £1 = 2.50 dollars £1 = $2.20 £1 = 13.25 kronor

Calculate (a) how many Italian euros £32.50 will buy, (b) the number of Canadian dollars that can be purchased for £74.80, (c) the pounds sterling which can be exchanged for 14 040 yen, (d) the pounds sterling which can be exchanged for 1754.30 Swedish kronor, and (e) the Australian dollars which can be bought for £55 [(a) 48.10 euros (b) $164.56 (c) £62.40 (d) £132.40 (e) 137.50 dollars]

2.54 cm = 1 inch 1.61 km = 1 mile

(a) At what time should a man catch a train at Mossley Hill to enable him to be in Manchester Piccadilly by 8.15 a.m.?

Further problems conversion tables and charts

Italy Japan Australia Canada Sweden

Length

Use the list to determine (a) the number of millimetres in 15 inches, (b) a speed of 35 mph in km/h, (c) the number of kilometres in 235 miles, (d) the number of pounds and ounces in 24 kg (correct to the nearest ounce), (e) the number of kilograms in 15 lb, (f ) the number of litres in 12 gallons and (g) the number of gallons in 25 litres. ⎡ ⎤ (a) 381 mm (b) 56.35 km/h ⎢ (c) 378.35 km (d) 52 lb 13 oz ⎥ ⎢ ⎥ ⎣ (e) 6.82 kg (f ) 54.55 litre ⎦ (g) 5.5 gallons

Thus 42 kg = 92 lb 6 oz, correct to the nearest ounce.

120 pints =

Below is a list of some metric to imperial conversions.

Capacity 1 litre = 1.76 pints (8 pints = 1 gallon)

30 kg = 13.64 kg 2.2

0.4 lb = 0.4 × 16 oz = 6.4 oz = 6 oz, correct to the nearest ounce

(f )

33

(c) A man living at Edge Hill has to be at work at Trafford Park by 8.45 a.m. It takes him 10 minutes to walk to his work from Trafford Park station. What time train should he catch from Edge Hill? ⎡ ⎤ (a) 7.09 a.m. ⎣ (b) 52 minutes, 31.15 m.p.h. ⎦ (c) 7.04 a.m.

4.4

Evaluation of formulae

The statement v = u + at is said to be a formula for v in terms of u, a and t . v, u, a and t are called symbols.

Section 1

Calculations and evaluation of formulae

Section 1

MX

MO

SX

SO

SX BHX

A Miles

♦

♦

C

C

SX BHX

BHX ♦

BHX

BHX

SX BHX

BHX

BHX

♦

♦

♦

D

E

C

BHX ♦

$$ I

I 05 25

I 05 37

06 03

I

06 23

06 30

I

06 54

07 00

07 17

07 30

I

Liverpool Lime Street

82, 99

d

1 12

Edge Hill

82, 99

d

06 34

07 04

07 34

07 52

08 00 08 04

08 23

08 30 08 34

3 12 4 12 5 12

Mossley Hill

82

d

06 39

07 09

07 39

08 09

08 39

West Allerton

82

d

06 41

07 11

07 41

08 11

08 41

Allerton

82

d

06 43

07 13

07 43

08 13

08 43

–

Liverpool Central

101

d

06 73

06 45

07 15

07 45

08 75

–

Garston (Merseyside)

101

d

06 26

06 56

07 26

07 56

08 26

07u07

08u05

7 12 8 12

Hunt’s Cross

d

06 17

06 47

07 17

07 47

Halewood

d

06 20

06 50

07 20

07 50

10 21 12 21

Hough Green

d

06 24

06 54

07 24

Widnes

d

06 27

06 57

07 27

16

Sankey for Penketh

d

00 02

06 32

07 02

07 32

18 21

Warrington Central

a

00 07

–

05u38

d

05u50

05 50

06 02

05 51

06 03

05 56

06 08

06 37 06 30

06 46 06 46

07 07 07 00

07 19 07 20

07 37 07 30

20 21

Padgate

d

21 21

Birchwood

d

24 21 25 21

Glazebrook

d

06 41

07 11

07 41

Iriam

d

06 02

06 44

07 14

07 44

28

Flixton

d

06 06

06 48

07 18

28 21

Chassen Road

d

06 08

06 50

07 20

29

Urmston

d

00 03

06 10

06 52

30 21

Humphrey Park

d

00 13

06 13

31

Trafford Park

d

06 15

34

Deansgate

81

d

00 23

34 21

Manchester Oxford Road

81

a

00 27

06 22

06 22

d

00 27

06 23

06 23

07 09

00 34

–

06 33 06 36

07 03 06 51

07 06

07 54 07 35

07 36

08 12

08 02 07 43 07 43

07 33 07 25

07 57

08 07 08 00

08 20

08 06

08 47

08 20

08 50

08 24

08 54

08 27

08 57

08 32

09 02

08 37

08 20

08 30

08 03 07 48

08 17

08 46 08 46

08 33 08 25

08 34

09 03 08 51

09 06

08 41

09 11

08 14

08 34

08 44

09 14

07 48

08 15

08 38

08 48

09 18

07 50

08 20

08 40

08 50

09 20

07 22

07 52

08 22

08 42

08 52

09 22

06 55

07 25

07 55

08 25

08 45

08 55

09 25

06 57

07 27

07 57

08 27

08 47

08 57

09 27

07 03 07 05

07 33 07 08

07 35

08 11

09 07 09 00

07 54

08 03

08 52

09 03

08 08

08 35

08 40

08 54

09 05

07 41

08 09

08 37

08 41

08 55

09 09

08 39

08 43

08 57

09 11

08 54

09 19

09 32

07 40

08 05

08 33

35

Manchester Piccadilly

81

a

06 25

06 25

07 11

07 43

08 11

–

Stockport

81, 90

a

06 34

06 34

07 32

07 54

08 32

–

Shetﬁeld

90

a

07 30

07 30

08 42

09 33 09 08

09 35

08 42 (Continued )

34 Engineering Mathematics

Table 4.3 Liverpool, Hunt’s Cross and Warrington → Manchester

Table 4.3 (Continued) BHX

Liverpool Lime Street

82, 99

d

Edge Hill

BHX

BHX

BHX

♦

♦

♦

♦

♦

♦

♦

♦

I

I

I

I

I

I

I

I

08 54

09 00

09 23

09 30

09 56

10 00

10 23

10 30

10 56

11 00

11 23

11 30

11 56

12 00

♦ I

12 23

12 30

12 56

13 00

82, 99

d

09 04

09 34

10 04

10 34

11 04

11 34

12 04

12 34

13 04

Mossley Hill

82

d

09 09

09 39

10 09

10 39

11 09

11 39

12 09

12 39

13 09

West Allerton

82

d

09 11

09 41

10 11

10 41

11 11

11 41

12 11

12 41

13 11

Allerton

82

d

09 13

09 43

10 13

10 43

11 13

11 43

12 13

12 43

Liverpool Central

101

d

09 45

Garston (Merseyside)

101

d

09 56

Hunt’s Cross

d

09u09

Halewood Hough Green

09 15

09 45

09 26

09 56

09 17

09 47

10u09

d

09 20

d

09 24

Widnes

d

09 27

Sankey for Penketh

d

Warrington Central

a

09 21

d

09 22

09 32 09 37 09 30

09 46

10 26

10 56

10 47

11u09

09 50

10 20

09 54

10 24

09 57

10 27

10 07 10 00

10 45

10 17

10 02 09 46

10 15

10 32 10 21 10 22

10 37

10 46

11 26

11 56

11 47

12u09

10 50

11 20

10 54

11 24

10 57

11 27

10 03

11 07 11 00

11 45

11 17

11 02 10 46

11 75

11 32 11 21 11 22

11 37

11 46

12 26

12 56

12 47

13u09

11 50

12 20

12 50

13 20

11 54

12 24

12 54

13 24

11 57

12 27

12 57

13 27

12 32 12 21 12 22

12 37

13 02 12 46 12 46

13 00

d

09 33

Birchwood

d

09 36

Glazebrook

d

09 41

10 11

11 11

12 11

13 11

Iriam

d

09 44

10 14

11 14

12 14

13 14

Flexton

d

09 48

10 18

11 18

12 18

13 18

Chassen Road

d

09 50

10 20

11 20

12 20

13 20

Urmston

d

09 52

10 22

11 22

12 22

13 22

Humphrey Park

d

09 55

10 25

11 25

12 25

13 25

Trafford Park

d

09 57

10 27

11 27

12 27

13 27

10 03

10 33

11 33

12 33

13 33

10 06

10 51

12 03

13 07

Padgate

09 51

11 03

12 07 12 00

11 06

11 51

12 45

12 17

12 02 11 46

13 13

12 15

13 32 13 21

13 03

12 06

12 51

13 04

d

Manchester Oxford Road

81

a

09 40

10 05

10 08

10 40

11 08

11 40

12 08

12 40

13 08

d

09 41

10 06

10 09

10 41

11 09

11 41

12 09

12 41

13 09

13 41

Manchester Piccadilly

12 35

13 35

13 40

81

a

09 43

10 08

10 11

10 43

11 11

11 43

12 11

12 43

13 11

13 43

Stockport

81, 90

a

09 54

10 25

10 32

10 54

11 32

11 54

12 32

12 54

13 32

13 54

Shefﬁeld

90

a

10 42

11 42

12 41

13 42

14 39

Calculations and evaluation of formulae

81

11 35

13 37

13 22

Deansgate

10 35

13 17

Reproduced with permission of British Rail

35

Section 1

Section 1

36 Engineering Mathematics The single term on the left-hand side of the equation, v, is called the subject of the formulae. Provided values are given for all the symbols in a formula except one, the remaining symbol can be made the subject of the formula and may be evaluated by using a calculator. Problem 16. In an electrical circuit the voltage V is given by Ohm’s law, i.e. V = IR. Find, correct to 4 signiﬁcant ﬁgures, the voltage when I = 5.36 A and R = 14.76 . V = IR = (5.36)(14.76) Hence, voltage V = 79.11 V, correct to 4 signiﬁcant ﬁgures. Problem 17. The surface area A of a hollow cone is given by A = πrl. Determine, correct to 1 decimal place, the surface area when r = 3.0 cm and l = 8.5 cm. A = πrl = π (3.0)(8.5) cm2 Hence, surface area A = 80.1 cm2 , correct to 1 decimal place.

The volume V cm3 of a right 1 circular cone is given by V = πr 2 h. Given that 3 r = 4.321 cm and h = 18.35 cm, ﬁnd the volume, correct to 4 signiﬁcant ﬁgures.

Problem 20.

1 1 V = πr 2 h = π(4.321)2 (18.35) 3 3 1 = π(18.671041)(18.35) 3 Hence volume,V = 358.8 cm3 , correct to 4 signiﬁcant ﬁgures. Problem 21.

Force F newtons is given by the Gm 1 m 2 formula F = , where m 1 and m 2 are d2 masses, d their distance apart and G is a constant. Find the value of the force given that G = 6.67 × 10−11, m 1 = 7.36, m 2 = 15.5 and d = 22.6. Express the answer in standard form, correct to 3 signiﬁcant ﬁgures.

F=

(6.67 × 10−11)(7.36)(15.5) Gm 1 m 2 = d2 (22.6)2

Problem 18. Velocity v is given by v = u + at . If u = 9.86 m/s, a = 4.25 m/s2 and t = 6.84 s, ﬁnd v, correct to 3 signiﬁcant ﬁgures. v = u + at = 9.86 + (4.25)(6.84)

=

(6.67)(7.36)(15.5) 1.490 = (1011 )(510.76) 1011

Hence force F = 1.49 × 10−11 newtons, correct to 3 signiﬁcant ﬁgures.

= 9.86 + 29.07 = 38.93 Hence, velocity v = 38.9 m/s, correct to 3 signiﬁcant ﬁgures. Problem 19. The power, P watts, dissipated in an electrical circuit may be expressed by the formula V2 . Evaluate the power, correct to 3 P= R signiﬁcant ﬁgures, given that V = 17.48 V and R = 36.12 . P=

V2 (17.48)2 305.5504 = = R 36.12 36.12

Hence power, P = 8.46 W, correct to 3 signiﬁcant ﬁgures.

Problem 22.

The time of swing t seconds, of a l simple pendulum is given by t = 2π g Determine the time, correct to 3 decimal places, given that l = 12.0 and g = 9.81 t = 2π

l = (2)π g

12.0 9.81

√ = (2)π 1.22324159 = (2)π(1.106002527) Hence time t = 6.950 seconds, correct to 3 decimal places.

Problem 23. Resistance, R, varies with temperature according to the formula R = R0 (1 + αt ). Evaluate R, correct to 3 signiﬁcant ﬁgures, given R0 = 14.59, α = 0.0043 and t = 80. R = R0 (1 + αt ) = 14.59[1 + (0.0043)(80)] = 14.59(1 + 0.344) = 14.59(1.344) Hence, resistance, R = 19.6 , correct to 3 signiﬁcant ﬁgures.

8. Evaluate resistance RT , given 1 1 1 1 = + + when R1 = 5.5 , RT R1 R2 R3 R2 = 7.42 and R3 = 12.6 [2.526 ] force × distance . Find the power 9. Power = time when a force of 3760 N raises an object a distance of 4.73 m in 35 s [508.1 W] 10. The potential difference, V volts, available at battery terminals is given by V = E − Ir. Evaluate V when E = 5.62, I = 0.70 and R = 4.30 [V = 2.61 V ] 11. Given force F = 12 m(v 2 − u 2 ), ﬁnd F when m = 18.3, v = 12.7 and u = 8.24 [F = 854.5]

Now try the following exercise Exercise 19 Further problems on evaluation of formulae 1. A formula used in connection with gases is R = (PV )/T . Evaluate R when P = 1500, V = 5 and T = 200 [R = 37.5] 2. The velocity of a body is given by v = u + at . The initial velocity u is measured when time t is 15 seconds and found to be 12 m/s. If the acceleration a is 9.81 m/s2 calculate the ﬁnal velocity v [159 m/s] 3. Find the distance s, given that s = gt 2, time t = 0.032 seconds and acceleration due to gravity g = 9.81 m/s2 [0.00502 m or 5.02 mm] 1 2

4. The energy stored in a capacitor is given by E = 12 CV 2 joules. Determine the energy when capacitance C = 5 × 10−6 farads and voltage V = 240V [0.144 J] 5. Resistance R2 is given by R2 = R1 (1 + αt ). Find R2 , correct to 4 signiﬁcant ﬁgures, when R1 = 220, α = 0.00027 and t = 75.6 [224.5] mass 6. Density = . Find the density when volume the mass is 2.462 kg and the volume is 173 cm3 . Give the answer in units of kg/m3 [14 230 kg/m3 ] 7. Velocity = frequency × wavelength. Find the velocity when the frequency is 1825 Hz and the wavelength is 0.154 m [281.1 m/s]

12. The current I amperes ﬂowing in a number nE of cells is given by I = . Evaluate R + nr the current when n = 36. E = 2.20, R = 2.80 and r = 0.50 [I = 3.81 A] 13. The time, t seconds, of oscillation for a siml . Deterple pendulum is given by t = 2π g mine the time when π = 3.142, l = 54.32 and g = 9.81 [t = 14.79 s] 14. Energy, E joules, is given by the formula E = 12 L I 2 . Evaluate the energy when L = 5.5 and I = 1.2 [E = 3.96 J] 15. The current I amperes in an a.c. circuit is V given by I = √ . Evaluate the cur2 R + X2 rent when V = 250, R = 11.0 and X = 16.2 [I = 12.77 A] 16. Distance s metres is given by the formula s = ut + 12 at 2 . If u = 9.50, t = 4.60 and a = −2.50, evaluate the distance [s = 17.25 m] 17. The area, √A, of any triangle is given by A = s(s − a)(s − b)(s − c) where a+b+c s= . Evaluate the area given 2 a = 3.60 cm, b = 4.00 cm and c = 5.20 cm [ A = 7.184 cm2 ] 18. Given that a = 0.290, b = 14.86, c = 0.042, d = 31.8and e = 0.650, evaluate v, given ab d that v = − [v = 7.327] c e

37

Section 1

Calculations and evaluation of formulae

Section 1

Revision Test 1 This Revision test covers the material contained in Chapters 1 to 4. The marks for each question are shown in brackets at the end of each question. 1.

2.

3.

1 2 Simplify (a) 2 ÷ 3 3 3 7 1 1 1 ÷ + +2 (b) 1 4 3 5 24 ×2 7 4

(b) 2.75 × 10−2 − 2.65 × 10−3 (9)

A piece of steel, 1.69 m long, is cut into three pieces in the ratio 2 to 5 to 6. Determine, in centimetres, the lengths of the three pieces (4) 576.29 19.3 (a) correct to 4 signiﬁcant ﬁgures (2)

5.

Express 54.7 mm as a percentage of 1.15 m, correct to 3 signiﬁcant ﬁgures (3)

6.

Evaluate the following:

8.

(14)

Express the following in both standard form and engineering notation: 2 (3) 5 Determine the value of the following, giving the answer in both standard form and engineering notation: (a) 1623 (b) 0.076 (c) 145

Convert the following binary numbers to decimal form: (a) 1101 (b) 101101.0101

(5)

10. Convert the following decimal number to binary form: (6)

(a) 479 (b) 185.2890625

(6)

12. Convert (a) 5F16 into its decimal equivalent (b) 13210 into its hexadecimal equivalent (c) 1101010112 into its hexadecimal equivalent (6) 13. Evaluate the following, each correct to 4 signiﬁcant ﬁgures: √ 1 (c) 0.0527 (a) 61.222 (b) (3) 0.0419

(a)

7.

(4)

11. Convert the following decimal numbers to binary, via octal:

Determine, correct to 1 decimal places, 57% of 17.64 g (2)

2 3 × 2 × 22 (23 × 16)2 (b) 4 2 (8 × 2)3 −1 1 1 (c) (d) (27)− 3 42 −2 2 3 − 2 9 (e) 2 2 3

9.

(a) 27 (b) 44.1875

Evaluate

(b) correct to 1 decimal place 4.

(a) 5.9 × 102 + 7.31 × 102

14. Evaluate the following, each correct to 2 decimal places: 3 36.22 × 0.561 (a) 27.8 × 12.83 14.692 (7) (b) √ 17.42 × 37.98 15. If 1.6 km = 1 mile, determine the speed of 45 miles/hour in kilometres per hour (3) 16. Evaluate B, correct to 3 signiﬁcant ﬁgures, when W = 7.20, υ = 10.0 and g = 9.81, given that W υ2 (3) B= 2g

Chapter 5

Algebra 5.1

Basic operations

Algebra is that part of mathematics in which the relations and properties of numbers are investigated by means of general symbols. For example, the area of a rectangle is found by multiplying the length by the breadth; this is expressed algebraically as A = l × b, where A represents the area, l the length and b the breadth. The basic laws introduced in arithmetic are generalised in algebra. Let a, b, c and d represent any four numbers. Then: (i) a + (b + c) = (a + b) +c

Replacing p, q and r with their numerical values gives: 3 3 2 3 2 1 4 p qr = 4(2) 2 2 1 3 3 3 = 4 × 2 × 2 × × × × = 27 2 2 2 2 Problem 3. Find the sum of: 3x, 2x, −x and −7x The sum of the positive term is: 3x + 2x = 5x The sum of the negative terms is: x + 7x = 8x Taking the sum of the negative terms from the sum of the positive terms gives: 5x − 8x = −3x

(ii) a(bc) = (ab)c Alternatively

(iii) a + b = b + a

3x + 2x + (−x) + (−7x) = 3x + 2x − x − 7x

(iv) ab =ba

= −3x

(v) a(b + c) = ab + ac a +b a b = + c c c (vii) (a + b)(c + d) = ac + ad + bc + bd (vi)

Problem 1. Evaluate: 3ab −2bc + abc when a = 1, b = 3 and c = 5 Replacing a, b and c with their numerical values gives: 3ab − 2bc + abc = 3 × 1 × 3 − 2 × 3 × 5 + 1×3×5 = 9 − 30 + 15 = −6 Problem 2. Find the value of 4 p2 qr 3, given the 1 1 p =2, q = and r = 1 2 2 DOI: 10.1016/B978-0-08-096562-8.00005-5

Problem 4. Find the sum of: 4a, 3b, c, −2a, −5b and 6c Each symbol must be dealt with individually. For the ‘a’ terms: +4a − 2a = 2a For the ‘b’ terms: +3b − 5b = −2b For the ‘c’ terms: +c + 6c = 7c Thus 4a + 3b + c + (−2a) + (−5b) + 6c = 4a + 3b + c − 2a − 5b + 6c = 2a − 2b + 7c Problem 5. Find the sum of: 5a − 2b, 2a + c, 4b − 5d and b − a + 3d − 4c

Section 1

40 Engineering Mathematics The algebraic expressions may be tabulated as shown below, forming columns for the a’s, b’s, c’s and d’s. Thus: +5a − 2b +2a +c + 4b − 5d −a + b − 4c + 3d Adding gives:

Problem 6. x − 2y + 5z

6a + 3b − 3c − 2d

Subtract 2x + 3y − 4z from

x − 2y + 5z 2x + 3y − 4z Subtracting gives: −x − 5y + 9z (Note that +5z −−4z =+5z + 4z =9z) An alternative method of subtracting algebraic expressions is to ‘change the signs of the bottom line and add’. Hence: x − 2y + 5z −2x − 3y + 4z

Multiplying by 2x → Multiplying

Multiply 2a + 3b by a + b

6x 2 − 4x y 2 + 8x 2 y

by −5y →

− 20x y 2

− 15x y + 10y 3

Adding gives: 6x2 − 24xy2 + 8x2 y − 15xy + 10y3 Problem 9.

Simplify: 2 p ÷ 8pq

2 p ÷8pq means

2p . This can be reduced by cancelling 8 pq

as in arithmetic. 1 2 × p1 2p 1 = = 8 pq 84 × p1 × q 4q

Thus:

Now try the following exercise Exercise 20

Further problems on basic operations

Find the value of 2xy + 3yz − xyz, when x = 2, y = −2 and z =4 [−16] 2 2. Evaluate 3pq 3r 3 when p = , q = −2 and 3 r = −1 [−8]

1.

3.

Find the sum of 3a, −2a, −6a, 5a and 4a [4a]

4.

Add together 2a + 3b + 4c, −5a − 2b + c, 4a − 5b − 6c [a − 4b − c]

5.

Add together 3d + 4e, −2e + f , 2d − 3 f , 4d − e + 2 f − 3e [9d − 2e]

Adding gives: −x − 5y + 9z

Problem 7.

3x − 2y 2 + 4x y 2x − 5y

Multiplying by a → 2a 2 + 3ab

From 4x − 3y + 2z subtract x + 2y − 3z [3x − 5y + 5z] 3 b b 7. Subtract a − + c from − 4a − 3c 2 3 2 1 5 −5 a + b − 4c 2 6 8. Multiply 3x + 2y by x − y [3x 2 − x y − 2y 2 ]

Multiplying by b →

+ 2ab + 3b 2

9.

Multiply 2a − 5b + c by 3a + b [6a 2 − 13ab + 3ac − 5b 2 + bc]

2a2 + 5ab + 3b2

10.

2 Simplify (i) 3a ÷ 9ab (ii) 4a b ÷ 2a 1 (i) (ii) 2ab 3b

Each term in the ﬁrst expression is multiplied by a, then each term in the ﬁrst expression is multiplied by b, and the two results are added. The usual layout is shown below. 2a + 3b a + b

Adding gives: Problem 8.

Multiply 3x − 2y 2 + 4xy by 2x − 5y

6.

5.2

Laws of indices

The laws of indices are: (i) a m × a n = a m+n

am = a m−n an √ (iv) a m/n = n a m (ii)

(iii) (a m )n = a mn 1 (v) a −n = n a

(vi) a 0 = 1

Problem 10. Simplify: a 3 b 2 c × ab3 c5 Grouping like terms gives: a 3 × a × b 2 × b3 × c × c5 Using the ﬁrst law of indices gives: a 3+1 × b2+3 × c1+5 a 4 × b5 × c6 = a4 b5 c6

i.e.

Problem 11. Simplify: a 1/2b 2 c−2 × a 1/6b 1/2c Using the ﬁrst law of indices, a 1/2b 2c−2 × a (1/6)b(1/2) c

p 1/2 q 2 r 2/3 and p 1/4 q 1/2r 1/6 evaluate when p =16, q = 9 and r = 4, taking positive roots only

Problem 13. Simplify:

Using the second law of indices gives: p (1/2)−(1/4)q 2−(1/2)r (2/3)−(1/6) = p1/4 q3/2 r1/2 When p = 16, q = 9 and r = 4, p1/4 q 3/2r 1/2 = (16)1/4 (9)3/2 (4)1/2 √ √ √ 4 = ( 16)( 93 )( 4) = (2)(33 )(2) = 108

Problem 14. Simplify:

a+b can be split c

x 2 y3 + x y2 x 2 y3 x y2 = + xy xy xy = x 2−1 y 3−1 + x 1−1 y 2−1 = xy2 + y

2/3 5/2 −1

b

x 2 y3 + x y2 xy

Algebraic expressions of the form a b into + . Thus c c

= a (1/2)+(1/6) × b 2+(1/2) × c−2+1 =a

41

c

(since x 0 = 1, from the sixth law of indices). a 3 b 2 c4 Problem 12. Simplify: and evaluate abc−2 1 when a = 3, b = and c = 2 8 Using the second law of indices, a3 b2 = a 3−1 = a 2 , = b 2−1 = b a b and

c4 = c4−(−2) = c6 c−2

a 3 b 2 c4 = a2 bc6 abc−2 1 When a = 3, b = and c = 2, 8 1 1 2 6 2 6 (2) = (9) (64) = 72 a bc = (3) 8 8

Problem 15. Simplify:

x2 y x y2 − x y

The highest common factor (HCF) of each of the three terms comprising the numerator and denominator is xy. Dividing each term by xy gives: x2 y x y2 − x y

=

x2 y xy x y2 xy

−

xy xy

=

x y−1

Thus

Problem 16. Simplify: ( p3 )1/2 (q 2 )4 Using the third law of indices gives: p3×(1/2)q 2×4 = p(3/2) q8

Section 1

Algebra

Section 1

42 Engineering Mathematics

Problem 17.

(mn 2 )3 Simplify: 1/2 1/4 4 (m n )

Using the second law of indices gives: d 2−3 e2−2 f (1/2)−5 = d −1 e0 f −9/2 = d −1 f (−9/2) since e0 = 1 from the sixth law of indices

The brackets indicate that each letter in the bracket must be raised to the power outside. Using the third law of indices gives: (mn 2 )3 m 1×3 n 2×3 m3 n6 = = ×4 (m 1/2 n 1/4)4 m2 n1 m (1/2)×4 n (1/4)

=

Problem 18. Simplify: √ √ √ √ 1 3 (a 3 b c5 )( a b 2 c3 ) and evaluate when a = , 4 b = 6 and c = 1 Using the fourth law of indices, the expression can be written as: (a b

3 1/2 5/2

c

)(a

df 9/2

from the ﬁfth law of indices.

Using the second law of indices gives: m3n 6 = m 3−2 n 6−1 = mn5 m2 n1

1

Problem 20.

√ (x 2 y 1/2 )( x 3 y 2 ) Simplify: (x 5 y 3 )1/2

Using the third and fourth laws of indices gives: √ (x 2 y 1/2 )( x 3 y 2 ) (x 2 y 1/2 )(x 1/2 y 2/3 ) = (x 5 y 3 )1/2 x 5/2 y 3/2 Using the ﬁrst and second laws of indices gives: x 2+(1/2)−(5/2) y (1/2)+(2/3)−(3/2) = x 0 y −1/3 = y −1/3

c )

1/2 2/3 3

b

or Using the ﬁrst law of indices gives: a 3+(1/2)b(1/2)+(2/3)c(5/2)+3 = a 7/2b 7/6c11/2 It is usual to express the answer in the same form as the question. Hence 6 a 7/2b 7/6c11/2 = a7 b7 c11 1 When a = , b = 64 and c = 1, 4 7 √ √ 1 6 6 7 7 11 a b c = 647 111 4 =

7 1 (2)7 (1) = 1 2

Now try the following exercise Exercise 21

d 2 e2 f 1/2 d 2 e2 f 1/2 = (d 3/2 e f 5/2 )2 d 3 e2 f 5

Further problems on laws of indices

1.

Simplify (x 2 y 3 z)(x 3 yz2 ) and evaluate when 1 x = , y = 2 and z = 3 2 1 5 4 3 x y z , 13 2

2.

Simplify (a 3/2bc−3 )(a 1/2bc−1/2 c) and evaluate when a = 3, b = 4 and c = 2 1 2 1/2 −2 a b c , ±4 2

3.

Simplify:

Simplify:

Using the third law of indices gives:

1 or √ 3 y

from the ﬁfth and sixth law of indices.

d 2 e2 f 1/2 expressing (d 3/2 e f 5/2)2 the answer with positive indices only

Problem 19.

1 y 1/3

a 5 bc3 3 and evaluate when a = , 2 3 2 a b c 2 2 1 [a 3 b −2c, 9] b = and c = 2 3

43

Collecting similar terms together gives: In Problems 4 to 10, simplify the given expressions: 4.

5.

6.

x 1/5 y 1/2 z 1/3 x −1/2 y 1/3 z −1/6

[x 7/10 y 1/6 z 1/2 ]

a2b + a3b a 2 b2

p 3q 2 pq 2 − p2 q

7.

(a 2 )1/2 (b 2 )3 (c1/2 )3

8.

(a 2 b−1 c−3 )3

9.

(abc)2

10.

(a b

p2q q−p

c )(ab) √ √ 3 ( a bc)

Problem 22. Simplify: a 2 − (2a − ab) − a(3b + a)

When the brackets are removed, both 2a and −ab in the ﬁrst bracket must be multiplied by −1 and both 3b and a in the second bracket by −a. Thus: a 2 − (2a − ab) − a(3b + a)

[ab 6 c3/2 ] [a −4 b 5 c11 ]

√ √ √ √ ( x y 3 3 z 2 )( x y 3 z 3 ) 3 1/2 −1/2

1+a b

3a + 3b − 2c − 4d

√ 6 [x y 3 z 13 ]

= a 2 − 2a + ab − 3ab − a 2 Collecting similar terms together gives: −2a − 2ab Since −2a is a common factor, the answer can be expressed as: −2a(1 +b) Problem 23. Simplify: (a + b)(a − b)

1/3

a 11/6 b 1/3c−3/2

√ 6 11 √ a 3b or √ c3

Each term in the second bracket has to be multiplied by each term in the ﬁrst bracket. Thus: (a + b)(a − b) = a(a − b) + b(a − b) = a 2 − ab + ab − b 2 = a2 − b 2

5.3

Brackets and factorisation

When two or more terms in an algebraic expression contain a common factor, then this factor can be shown outside of a bracket. For example ab + ac = a(b + c) which is simply the reverse of law (v) of algebra on page 34, and 6 px + 2 py − 4 pz = 2 p(3x + y − 2z)

Alternatively

Multiplying by a → a 2 + ab Multiplying by −b → − ab − b 2 Adding gives:

(3a + b) + 2(b + c) − 4(c + d) = 3a + b + 2b + 2c − 4c − 4d

− b2

(2x − 3y)2 = (2x − 3y)(2x − 3y) = 2x(2x − 3y) − 3y(2x − 3y) = 4x 2 − 6x y − 6x y + 9y 2

Problem 21. Remove the brackets and simplify the expression:

Both b and c in the second bracket have to be multiplied by 2, and c and d in the third bracket by −4 when the brackets are removed. Thus:

a2

Problem 24. Simplify: (3x − 3y)2

This process is called factorisation.

(3a + b) + 2(b + c) − 4(c + d)

a + b a − b

= 4x2 − 12xy + 9y2 Alternatively,

2x − 3y 2x − 3y

Multiplying by 2x → 4x 2 − 6x y Multiplying by −3y → − 6x y + 9y 2 Adding gives:

4x 2 − 12x y + 9y 2

Section 1

Algebra

Section 1

44 Engineering Mathematics Problem 25. expression:

Remove the brackets from the 2[ p 2 − 3(q +r) + q 2 ]

For each part of this problem, the HCF of the terms will become one of the factors. Thus: (a) xy − 3xz = x( y − 3z)

In this problem there are two brackets and the ‘inner’ one is removed ﬁrst.

(b) 4a 2 + 16ab3 = 4a(a +4b3 )

Hence, 2[ p 2 − 3(q +r) + q 2]

(c) 3a 2 b − 6ab2 + 15ab =3ab(a −2b + 5)

= 2[ p 2 − 3q − 3r + q 2 ] = 2p2 − 6q − 6r + 2q2 Problem 26. Remove the brackets and simplify the expression: 2a − [3{2(4a − b) − 5(a + 2b)} + 4a] Removing the innermost brackets gives: 2a − [3{8a − 2b − 5a − 10b} + 4a] Collecting together similar terms gives: 2a − [3{3a − 12b} + 4a]

Problem 29.

Factorise: ax − ay + bx − by

The ﬁrst two terms have a common factor of a and the last two terms a common factor of b. Thus: ax − ay + bx − by = a(x − y) + b(x − y) The two newly formed terms have a common factor of (x − y). Thus: a(x − y) + b(x − y) = (x − y)(a + b) Problem 30.

Factorise: 2ax − 3ay + 2bx − 3by

Removing the ‘curly’ brackets gives: 2a − [9a − 36b + 4a] Collecting together similar terms gives:

a is a common factor of the ﬁrst two terms and b a common factor of the last two terms. Thus: 2ax − 3ay + 2bx − 3by

2a − [13a − 36b]

= a(2x − 3y) + b(2x − 3y)

Removing the outer brackets gives: 2a − 13a − 36b

(2x − 3y) is now a common factor, thus:

i.e. −11a + 36b or 36b −11a

a(2x − 3y) + b(2x − 3y)

(see law (iii), page 39) Problem 27.

Simplify: x(2x − 4y) − 2x(4x + y)

= (2x − 3y)(a + b) Alternatively, 2x is a common factor of the original ﬁrst and third terms and −3y is a common factor of the second and fourth terms. Thus:

Removing brackets gives:

2ax − 3ay + 2bx − 3by

2x 2 − 4x y − 8x 2 − 2x y Collecting together similar terms gives: −6x 2 − 6x y Factorising gives: −6x(x + y) (since −6x is common to both terms). Problem 28. Factorise: (a) xy − 3xz (c) 3a 2 b − 6ab2 + 15ab (b) 4a 2 + 16ab3

= 2x(a + b) − 3y(a + b) (a + b) is now a common factor thus: 2x(a + b) − 3y(a + b) = (a + b)(2x − 3y) as before. Problem 31.

Factorise: x 3 + 3x 2 − x − 3

x 2 is a common factor of the ﬁrst two terms, thus: x 3 + 3x 2 − x − 3 = x 2 (x + 3) − x − 3

−1 is a common factor of the last two terms, thus: x 2 (x + 3) − x − 3 = x 2 (x + 3) − 1(x + 3) (x + 3) is now a common factor, thus: x 2 (x + 3) − 1(x + 3) = (x + 3)(x2 − 1) Now try the following exercise Exercise 22

Further problems on brackets and factorisation

In Problems 1 to 9, remove the brackets and simplify where possible: 1. (x + 2y) + (2x − y)

[3x + y]

2. 2(x − y) − 3( y − x)

[5(x − y)]

3. 2( p +3q −r) − 4(r − q + 2 p) + p [−5 p + 10q − 6r] 4. (a + b)(a + 2b)

Fundamental laws and precedence

The laws of precedence which apply to arithmetic also apply to algebraic expressions. The order is Brackets, Of, Division, Multiplication, Addition and Subtraction (i.e. BODMAS). Problem 32. Simplify: 2a + 5a × 3a − a Multiplication is performed before addition and subtraction thus: 2a + 5a × 3a − a = 2a + 15a 2 − a = a + 15a2 or a(1 + 15a) Problem 33. Simplify: (a + 5a) × 2a − 3a The order of precedence is brackets, multiplication, then subtraction. Hence (a + 5a) × 2a − 3a = 6a × 2a − 3a

[a 2 + 3ab +2b 2 ]

= 12a 2 − 3a or 3a(4a − 1)

[3 p2 + pq − 2q 2 ]

5. ( p + q)(3 p −2q)

2 6. (i) (x − 2y)2 (ii) (3a − b) (i) x 2 − 4x y + 4y 2

(ii) 7. 3a + 2[a − (3a − 2)] 8.

5.4

2 − 5[a(a − 2b) −(a − b)2 ]

9a 2 − 6ab +b2 [4 −a] [2 + 5b 2 ]

9. 24 p −[2{3(5 p − q) − 2( p + 2q)} + 3q] [11q − 2 p] In Problems 10 to 12, factorise: 10. (i) pb + 2 pc (ii) 2q 2 + 8qn [(i) p(b + 2c) (ii) 2q(q + 4n)] 11. (i) 21a 2 b2 − 28ab (ii) 2xy2 + 6x 2 y + 8x 3 y (i) 7ab(3ab −4) (ii) 2x y(y + 3x + 4x 2 12. (i) ay + by + a + b (ii) px + q x + py + q y (iii) 2ax + 3ay − 4bx − 6by ⎡ ⎤ (i) (a + b)(y + 1) ⎢ ⎥ ⎣ (ii) ( p + q)(x + y) ⎦ (iii) (a − 2b)(2x + 3y)

45

Problem 34. Simplify: a + 5a × (2a − 3a) The order of precedence is brackets, multiplication, then subtraction. Hence a + 5a × (2a − 3a) = a + 5a × −a = a + −5a 2 = a − 5a 2 or a(1 − 5a) Problem 35. Simplify: a ÷ 5a + 2a − 3a The order of precedence is division, then addition and subtraction. Hence a + 2a − 3a a ÷ 5a + 2a − 3a = 5a 1 1 = + 2a − 3a = − a 5 5 Problem 36. Simplify: a ÷ (5a + 2a) − 3a The order of precedence is brackets, division and subtraction. Hence a ÷ (5a + 2a) − 3a = a ÷ 7a − 3a a 1 = − 3a = − 3a 7a 7

Section 1

Algebra

Section 1

46 Engineering Mathematics Problem 37.

Simplify:

Problem 40.

Simplify:

3c + 2c × 4c + c ÷ 5c − 8c

(2a − 3) ÷ 4a + 5 × 6 − 3a

The order of precedence is division, multiplication, addition and subtraction. Hence:

The bracket around the (2a − 3) shows that both 2a and −3 have to be divided by 4a, and to remove the bracket the expression is written in fraction form.

3c + 2c × 4c + c ÷ 5c − 8c c − 8c = 3c + 2c × 4c + 5c 1 = 3c + 8c2 + − 8c 5 1 1 2 = 8c − 5c + or c(8c − 5) + 5 5 Problem 38.

(2a − 3) ÷ 4a + 5 × 6 − 3a

Hence,

2a − 3 + 5 × 6 − 3a 4a 2a − 3 = + 30 − 3a 4a 3 2a − + 30 − 3a = 4a 4a 1 3 = − + 30 − 3a 2 4a 1 3 = 30 − − 3a 2 4a =

Simplify:

3c + 2c × 4c + c ÷ (5c − 8c) The order of precedence is brackets, division, multiplication and addition. Hence, 3c + 2c × 4c + c ÷ (5c − 8c) = 3c + 2c × 4c + c ÷ −3c c = 3c + 2c × 4c + −3c c 1 Now = −3c −3 Multiplying numerator and denominator by −1 gives: 1 × −1 −3 × −1

i.e. −

1 3

Problem 41.

1 of 3 p + 4 p(3 p − p) 3 Applying BODMAS, the expression becomes 1 of 3 p + 4 p × 2 p 3 and changing ‘of’ to ‘×’ gives: 1 × 3p +4p × 2p 3

Hence: 3c + 2c × 4c +

c −3c

1 = 3c + 2c × 4c − 3 1 1 = 3c + 8c2 − or c(3 + 8c) − 3 3 Problem 39.

Simplify:

(3c + 2c)(4c + c) ÷ (5c − 8c)

Simplify:

i.e. p +8p2 or p(1 +8p) Now try the following exercise Exercise 23 Further problems on fundamental laws and precedence Simplify the following:

The order of precedence is brackets, division and multiplication. Hence (3c + 2c)(4c + c) ÷ (5c − 8c) 5c = 5c × 5c ÷ −3c = 5c × −3c 25 5 = 5c × − = − c 3 3

1.

2x ÷ 4x + 6x

2.

2x ÷ (4x + 6x)

3.

3a − 2a × 4a + a

1 + 6x 2 1 5

[4a(1 −2a)]

4. 3a − 2a(4a + a) 5. 2y + 4 ÷6y + 3 × 4 −5y 6. 2y + 4 ÷6y + 3(4 − 5y) 7. 3 ÷ y + 2 ÷ y + 1 8.

p 2 − 3 pq × 2 p ÷6q + pq

9. (x + 1)(x − 4) ÷ (2x + 2) 10.

5.5

1 of 2y + 3y(2y − y) 4

[a(3 −10a)] 2 − 3y + 12 3y 2 + 12 −13y 3y 5 +1 y [ pq] 1 (x − 4) 2 1 y + 3y 2

Direct and inverse proportionality

An expression such as y = 3x contains two variables. For every value of x there is a corresponding value of y. The variable x is called the independent variable and y is called the dependent variable. When an increase or decrease in an independent variable leads to an increase or decrease of the same proportion in the dependent variable this is termed direct proportion. If y = 3x then y is directly proportional to x, which may be written as y α x or y = kx, where k is called the coefﬁcient of proportionality (in this case, k being equal to 3). When an increase in an independent variable leads to a decrease of the same proportion in the dependent variable (or vice versa) this is termed inverse proportion. 1 If y is inversely proportional to x then y α or y = k/x. x Alternatively, k = x y, that is, for inverse proportionality the product of the variable is constant. Examples of laws involving direct and inverse proportional in science include: (i) Hooke’s law, which states that within the elastic limit of a material, the strain ε produced is directly proportional to the stress, σ , producing it, i.e. ε α σ or ε = kσ . (ii) Charles’s law, which states that for a given mass of gas at constant pressure the volume V is directly

47

proportional to its thermodynamic temperature T , i.e. V α T or V = kT . (iii) Ohm’s law, which states that the current I ﬂowing through a ﬁxed resistor is directly proportional to the applied voltage V , i.e. I α V or I = kV. (iv) Boyle’s law, which states that for a gas at constant temperature, the volume V of a ﬁxed mass of a gas is inversely proportional to its absolute pressure p, i.e. p α (1/ V ) or p =k/ V , i.e. pV = k. Problem 42. If y is directly proportional to x and y = 2.48 when x = 0.4, determine (a) the coefﬁcient of proportionality and (b) the value of y when x = 0.65 (a)

y α x, i.e. y = kx. If y = 2.48 when x = 0.4, 2.48 = k(0.4) Hence the coefﬁcient of proportionality, 2.48 k= = 6.2 0.4

(b)

y = kx, hence, when x = 0.65, y = (6.2)(0.65) =4.03

Problem 43. Hooke’s law states that stress σ is directly proportional to strain ε within the elastic limit of a material. When, for mild steel, the stress is 25 ×106 Pascals, the strain is 0.000125. Determine (a) the coefﬁcient of proportionality and (b) the value of strain when the stress is 18 ×106 Pascals (a)

σ α ε, i.e. σ = kε, from which k = σ/ε. Hence the coefﬁcient of proportionality, k=

25 × 106 = 200 × 109 pascals 0.000125

(The coefﬁcient of proportionality k in this case is called Young’s Modulus of Elasticity.) (b) Since σ = kε, ε = σ /k Hence when σ = 18 ×106 , strain ε =

18 × 106 = 0.00009 200 × 109

Problem 44. The electrical resistance R of a piece of wire is inversely proportional to the crosssectional area A. When A = 5 mm2 , R = 7.02 ohms. Determine (a) the coefﬁcient of proportionality and (b) the cross-sectional area when the resistance is 4 ohms

Section 1

Algebra

Section 1

48 Engineering Mathematics (a)

1 , i.e. R = k/ A or k = R A. Hence, when A R = 7.2 and A = 5, the coefﬁcient of proportionality, k = (7.2)(5) =36 Rα

(b) Since k = R A then A = k/R When R = 4, the cross-sectional area, A=

36 = 9 mm2 4

Problem 45. Boyle’s law states that at constant temperature, the volume V of a ﬁxed mass of gas is inversely proportional to its absolute pressure p. If a gas occupies a volume of 0.08 m3 at a pressure of 1.5 × 106 Pascals determine (a) the coefﬁcient of proportionality and (b) the volume if the pressure is changed to 4 ×106 Pascals (a)

1 i.e. V = k/ p or k = pV p Hence the coefﬁcient of proportionality, Vα

k = (1.5 × 106 )(0.08) = 0.12 × 106 (b) Volume V =

k 0.12 × 106 = 0.03 m3 = p 4 ×106

Now try the following exercise Exercise 24

Further problems on direct and inverse proportionality

1. If p is directly proportional to q and p = 37.5 when q = 2.5, determine (a) the constant of proportionality and (b) the value of p when q is 5.2 [(a) 15 (b) 78]

2. Charles’s law states that for a given mass of gas at constant pressure the volume is directly proportional to its thermodynamic temperature. A gas occupies a volume of 2.25 litres at 300 K. Determine (a) the constant of proportionality, (b) the volume at 420 K, and (c) the temperature when the volume is 2.625 litres [(a) 0.0075 (b) 3.15 litres (c) 350 K] 3. Ohm’s law states that the current ﬂowing in a ﬁxed resistor is directly proportional to the applied voltage. When 30 volts is applied across a resistor the current ﬂowing through the resistor is 2.4 ×10−3 amperes. Determine (a) the constant of proportionality, (b) the current when the voltage is 52 volts and (c) the voltage when the current is 3.6 × 10−3 amperes (a) 0.00008 (b) 4.16 × 10−3 A (c) 45 V 4. If y is inversely proportional to x and y = 15.3 when x = 0.6, determine (a) the coefﬁcient of proportionality, (b) the value of y when x is 1.5, and (c) the value of x when y is 27.2 [(a) 9.18 (b) 6.12 (c) 0.3375] 5. Boyle’s law states that for a gas at constant temperature, the volume of a ﬁxed mass of gas is inversely proportional to its absolute pressure. If a gas occupies a volume of 1.5 m3 at a pressure of 200 ×103 Pascals, determine (a) the constant of proportionality, (b) the volume when the pressure is 800 ×103 Pascals and 3 (c) the pressure when the volume is 1.25 m 3 3 (b) 0.375 m (a) 300 ×10 (c) 240 ×103 Pa

Chapter 6

Further algebra 6.1

Polynomial division

Before looking at long division in algebra let us revise long division with numbers (we may have forgotten, since calculators do the job for us!) 208 For example, is achieved as follows: 16 13 16 208 16 __ 48 48 __ .. __ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

16 divided into 2 won’t go 16 divided into 20 goes 1 Put 1 above the zero Multiply 16 by 1 giving 16 Subtract 16 from 20 giving 4 Bring down the 8 16 divided into 48 goes 3 times Put the 3 above the 8 3 × 16 = 48 48 − 48 = 0 Hence

Similarly,

208 = 13 exactly 16

Below are some examples of division in algebra, which in some respects, is similar to long division with numbers. (Note that a polynomial is an expression of the form f (x) = a + bx + cx 2 + d x 3 + · · · and polynomial division is sometimes required when resolving into partial fractions — see Chapter 7). Problem 1. Divide 2x 2 + x − 3 by x − 1 2x 2 + x − 3 is called the dividend and x − 1 the divisor. The usual layout is shown below with the dividend and divisor both arranged in descending powers of the symbols. 2x + 3 x − 1 2x 2 + x − 3 2x 2 − 2x _______

3x − 3 3x −3 _____ . . _____

22 15 __ 7 __

Thus (2x2 + x − 3) ÷ (x − 1) = (2x + 3)

11

15 172 15 __

DOI: 10.1016/B978-0-08-096562-8.00006-7

175 7 7 = 11 remainder 7 or 11 + = 11 15 15 15

Dividing the ﬁrst term of the dividend by the ﬁrst term 2x 2 of the divisor, i.e. gives 2x, which is put above the x ﬁrst term of the dividend as shown. The divisor is then multiplied by 2x, i.e. 2x(x − 1) = 2x 2 − 2x, which is placed under the dividend as shown. Subtracting gives 3x − 3. The process is then repeated, i.e. the ﬁrst term of the divisor, x, is divided into 3x, giving +3, which is placed above the dividend as shown. Then 3(x − 1) = 3x − 3 which is placed under the 3x − 3. The remainder, on subtraction, is zero, which completes the process.

172 is laid out as follows: 15

Hence

Section 1

50 Engineering Mathematics [A check can be made on this answer by multiplying (2x + 3) by (x − 1) which equals 2x 2 + x − 3] Problem 2.

(7) x into xy2 goes y 2 . Put y 2 above divided (8) y 2 (x + y) = x y 2 + y 3 (9) Subtract

Divide 3x 3 + x 2 + 3x + 5 by x + 1 Thus (1) (4) (7) 2 3x − 2x + 5 x + 1 3x 3 + x 2 + 3x + 5 3 + 3x 2 3x ________

−2x 2 + 3x + 5 2 − 2x −2x _________

The zero’s shown in the dividend are not normally shown, but are included to clarify the subtraction process and to keep similar terms in their respective columns. Problem 4.

(5) (6) (7) (8) (9)

x into 3x 3 goes 3x 2 . Put 3x 2 above 3x 3 3x 2 (x + 1) = 3x 3 + 3x 2 Subtract x into −2x 2 goes −2x. Put −2x above the dividend −2x(x + 1) = −2x 2 − 2x Subtract x into 5x goes 5. Put 5 above the dividend 5(x + 1) = 5x + 5 Subtract

5x − 2 5x − 10 ______ ______8 Hence

−2ab2 + 5b 3 2 + b3 −2ab ___________ 4b 3 ___________

+ y3

+ 2 y − x y2 −x __________ x y2 + y3 x y2 + y3 _______ . . _______ (1) (2) (3) (4) (5) (6)

+ 5b3

−4a 2 b + 5b3 −4a 2 b + 2ab 2 ____________

2

−x 2 y

2a 2 − 2ab − b 2

2a − b 4a 3 − 6a 2 b 3 − 2a 2 b 4a _________

(4) (7)

2 x − xy + y x + y x3 + 0 + 0 3 + x2 y x________

Divide 4a 3 − 6a 2 b + 5b 3 by 2a − b

x 3 + y3 Simplify: x+y (1)

x 2 + 3x − 2 8 = x+5+ x −2 x−2

Problem 5.

3x 3 + x 2 + 3x + 5 Thus = 3x2 − 2x + 5 x +1

Problem 3.

Divide (x 2 + 3x − 2) by (x − 2) x +5 x − 2 x 2 + 3x − 2 x 2 − 2x ______

5x + 5 5x + 5 ______ . . ______ (1) (2) (3) (4)

x 3 + y3 = x2 − xy + y2 x+y

Thus 4a 3 − 6a 2 b + 5b 3 4b3 = 2a2 − 2ab − b2 + 2a − b 2a − b

y3

x into x 3 goes x 2 . Put x 2 above x 3 of dividend x 2 (x + y) = x 3 + x 2 y Subtract x into −x 2 y goes −x y. Put −x y above dividend −x y(x + y) = −x 2 y − x y 2 Subtract

Now try the following exercise Exercise 25 Further problems on polynomial division 1.

Divide (2x 2 + x y − y 2 ) by (x + y)

[2x − y]

2.

Divide (3x 2 + 5x − 2) by (x + 2)

[3x − 1]

3.

Determine (10x 2 + 11x − 6) ÷ (2x + 3) [5x − 2]

4.

Find:

5.

Divide (x 3 + 3x 2 y + 3x y 2 + y3 ) by (x + y) [x 2 + 2x y + y 2 ]

6.

Find (5x 2 − x + 4) ÷ (x − 1)

7.

14x 2 − 19x − 3 2x − 3

[7x + 1]

8 5x + 4 + x −1

Divide (3x 3 + 2x 2 − 5x + 4) by (x + 2)

2 3x − 4x + 3 − x +2

3

8.

5x 4 + 3x 3 − 2x + 1 Determine: x −3 481 3 5x + 18x 2 + 54x + 160 + x −3

51

we could deduce at once that (x − 2) is a factor of the expression x 2 + 2x − 8. We wouldn’t normally solve quadratic equations this way — but suppose we have to factorise a cubic expression (i.e. one in which the highest power of the variable is 3). A cubic equation might have three simple linear factors and the difﬁculty of discovering all these factors by trial and error would be considerable. It is to deal with this kind of case that we use the factor theorem. This is just a generalised version of what we established above for the quadratic expression. The factor theorem provides a method of factorising any polynomial, f (x), which has simple factors. A statement of the factor theorem says: ‘if x = a is a root of the equation f (x) = 0, then (x − a) is a factor of f (x)’ The following worked problems show the use of the factor theorem. Problem 6. Factorise: x 3 − 7x − 6 and use it to solve the cubic equation: x 3 − 7x − 6 = 0 Let f (x) = x 3 − 7x − 6

6.2

The factor theorem

There is a simple relationship between the factors of a quadratic expression and the roots of the equation obtained by equating the expression to zero. For example, consider the quadratic equation x 2 + 2x − 8 = 0 To solve this we may factorise the quadratic expression x 2 + 2x − 8 giving (x − 2)(x + 4) Hence (x − 2)(x + 4) = 0 Then, if the product of two number is zero, one or both of those numbers must equal zero. Therefore,

If x = 1, then f (1) = 13 − 7(1) − 6 = −12 If x = 2, then f (2) = 23 − 7(2) − 6 = −12 If x = 3, then f (3) = 33 − 7(3) − 6 = 0 If f (3) = 0, then (x − 3) is a factor — from the factor theorem. We have a choice now. We can divide x 3 − 7x − 6 by (x − 3) or we could continue our ‘trial and error’ by substituting further values for x in the given expression — and hope to arrive at f (x) = 0. Let us do both ways. Firstly, dividing out gives:

either (x − 2) = 0, from which, x = 2 or (x + 4) = 0, from which, x = −4

2 x + 3x + 2 x − 3 x 3 + 0 − 7x − 6 3 − 3x 2 x_______

It is clear then that a factor of (x − 2) indicates a root of +2, while a factor of (x + 4) indicates a root of −4. In general, we can therefore say that:

3x 2 − 7x − 6 2 − 9x 3x ___________

a factor of (x − a) corresponds to a root of x = a In practice, we always deduce the roots of a simple quadratic equation from the factors of the quadratic expression, as in the above example. However, we could reverse this process. If, by trial and error, we could determine that x = 2 is a root of the equation x 2 + 2x − 8 = 0

2x − 6 2x −6 ______ . . ______ Hence

x 3 − 7x − 6 = x 2 + 3x + 2 x −3

Section 1

Further algebra

Section 1

52 Engineering Mathematics i.e. x 3 − 7x − 6 = (x − 3)(x 2 + 3x + 2) x 3 + 3x + 2 factorises ‘on sight’ as (x + 1)(x + 2) Therefore

Alternatively, having obtained one factor, i.e. (x − 1) we could divide this into (x 3 − 2x 2 − 5x + 6) as follows: 2 x −x −6 x − 1 x 3 − 2x 2 − 5x + 6 3 − x2 x_______

x3 − 7x − 6 = (x − 3)(x + 1)(x + 2) A second method is to continue to substitute values of x into f (x). Our expression for f (3) was 33 − 7(3) − 6. We can see that if we continue with positive values of x the ﬁrst term will predominate such that f (x) will not be zero. Therefore let us try some negative values for x: f (−1) = (−1)3 − 7(−1) − 6 = 0; hence (x + 1) is a factor (as shown above). Also, f (−2) = (−2)3 − 7(−2) − 6 = 0; hence (x + 2) is a factor (also as shown above). To solve x 3 − 7x − 6 = 0, we substitute the factors, i.e.

− x 2 − 5x + 6 2 + x − x___________ −6x + 6 −6x +6 ______ . . ______ Hence x 3 − 2x 2 − 5x + 6 = (x − 1)(x 2 − x − 6) = (x − 1)(x − 3)(x + 2)

(x − 3)(x + 1)(x + 2) = 0 from which, x = 3, x = −1 and x = −2 Note that the values of x, i.e. 3, −1 and −2, are all factors of the constant term, i.e. the 6. This can give us a clue as to what values of x we should consider.

Summarising, the factor theorem provides us with a method of factorising simple expressions, and an alternative, in certain circ*mstances, to polynomial division. Now try the following exercise

Problem 7. Solve the cubic equation x 3 − 2x 2 − 5x + 6 = 0 by using the factor theorem Let f (x) = x 3 − 2x 2 − 5x + 6 and let us substitute simple values of x like 1, 2, 3, −1, −2, and so on. f (1) = 13 − 2(1)2 − 5(1) + 6 = 0, hence (x − 1) is a factor f (2) = 23 − 2(2)2 − 5(2) + 6 = 0 f (3) = 33 − 2(3)2 − 5(3) + 6 = 0, hence (x − 3) is a factor f (−1) = (−1)3 − 2(−1)2 − 5(−1) + 6 = 0 f (−2) = (−2)3 − 2(−2)2 − 5(−2) + 6 = 0, hence (x + 2) is a factor Hence, x 3 − 2x 2 − 5x + 6 = (x − 1)(x − 3)(x + 2) Therefore if x 3 − 2x 2 − 5x + 6 = 0 then

(x − 1)(x − 3)(x + 2) = 0

from which, x = 1, x = 3 and x = −2

Exercise 26

Further problems on the factor theorem

Use the factor theorem to factorise the expressions given in problems 1 to 4. 1.

x 2 + 2x − 3

2.

x 3 + x 2 − 4x − 4

3. 2x 3 + 5x 2 − 4x − 7

[(x − 1)(x + 3)] [(x + 1)(x + 2)(x − 2)] [(x + 1)(2x 2 + 3x − 7)]

4. 2x 3 − x 2 − 16x + 15 [(x − 1)(x + 3)(2x − 5)] 5. Use the factor theorem to factorise x 3 + 4x 2 + x − 6 and hence solve the cubic equation x 3 + 4x 2 + x − 6 = 0 ⎤ ⎡ 3 x + 4x 2 + x − 6 ⎣ = (x − 1)(x + 3)(x + 2); ⎦ x = 1, x = −3 and x = −2 6. Solve the equation x 3 − 2x 2 − x + 2 = 0 [x = 1, x = 2 and x = −1]

6.3

53

In this case the other factor is (2x + 3), i.e.

The remainder theorem (ax2 + bx + c)

Dividing a general quadratic expression by (x − p), where p is any whole number, by long division (see Section 6.1) gives: ax + (b + ap) x − p ax 2 + bx +c 2 − apx ax ________ (b + ap)x + c (b + ap)x − (b + ap) p __________________ c + (b + ap) p __________________ The remainder, c + (b + ap) p = c + bp + ap2 or ap 2 + bp + c. This is, in fact, what the remainder theorem states, i.e. ‘if (ax2 + bx + c) is divided by (x − p), the remainder will be ap2 + bp + c’ If, in the dividend (ax 2 + bx + c), we substitute p for x we get the remainder ap 2 + bp + c For example, when (3x 2 − 4x + 5) is divided by (x − 2) the remainder is ap2 + bp + c, (where a = 3, b = −4, c = 5 and p = 2), i.e. the remainder is: 3(2)2 + (−4)(2) + 5 = 12 − 8 + 5 = 9 We can check this by dividing (3x 2 − 4x + 5) by (x − 2) by long division: 3x + 2 x − 2 3x 2 − 4x + 5 2 − 6x 3x _______

2x + 5 2x −4 ______ 9 ______ Similarly, when (4x 2 − 7x + 9) is divided by (x + 3), the remainder is ap 2 + bp + c, (where a = 4, b = −7, c = 9 and p = −3) i.e. the remainder is: 4(−3)2 + (−7)(−3) + 9 = 36 + 21 + 9 = 66 Also, when (x 2 + 3x − 2) is divided by (x − 1), the remainder is 1(1)2 + 3(1) − 2 = 2 It is not particularly useful, on its own, to know the remainder of an algebraic division. However, if the remainder should be zero then (x − p) is a factor. This is very useful therefore when factorising expressions. For example, when (2x 2 + x − 3) is divided by (x − 1), the remainder is 2(1)2 + 1(1) − 3 = 0, which means that (x − 1) is a factor of (2x 2 + x − 3).

(2x 2 + x − 3) = (x − 1)(2x − 3). The remainder theorem may also be stated for a cubic equation as: ‘if (ax3 + bx2 + cx + d) is divided by (x − p), the remainder will be ap3 + bp2 + cp + d’ As before, the remainder may be obtained by substituting p for x in the dividend. For example, when (3x 3 + 2x 2 − x + 4) is divided by (x − 1), the remainder is: ap3 + bp 2 + cp + d (where a = 3, b = 2, c = −1, d = 4 and p = 1), i.e. the remainder is: 3(1)3 + 2(1)2 + (−1)(1) + 4 = 3 + 2 − 1 + 4 = 8. Similarly, when (x 3 − 7x − 6) is divided by (x − 3), the remainder is: 1(3)3 + 0(3)2 − 7(3) − 6 = 0, which mean that (x − 3) is a factor of (x 3 − 7x − 6). Here are some more examples on the remainder theorem. Problem 8. Without dividing out, ﬁnd the remainder when 2x 2 − 3x + 4 is divided by (x − 2) By the remainder theorem, the remainder is given by: ap 2 + bp + c, where a = 2, b = −3, c = 4 and p = 2. Hence the remainder is: 2(2)2 + (−3)(2) + 4 = 8 − 6 + 4 = 6 Problem 9. Use the remainder theorem to determine the remainder when (3x 3 − 2x 2 + x − 5) is divided by (x + 2) By the remainder theorem, the remainder is given by: ap 3 + bp2 + cp + d, where a = 3, b = −2, c = 1, d = −5 and p = −2 Hence the remainder is: 3(−2)3 + (−2)(−2)2 + (1)(−2) + (−5) = −24 − 8 − 2 − 5 = −39 Problem 10. Determine the remainder when (x 3 − 2x 2 − 5x + 6) is divided by (a) (x − 1) and (b) (x + 2). Hence factorise the cubic expression

Section 1

Further algebra

Section 1

54 Engineering Mathematics (a) When (x 3 − 2x 2 − 5x + 6) is divided by (x − 1), the remainder is given by ap3 + bp2 + cp + d, where a = 1, b = −2, c = −5, d = 6 and p = 1, i.e. the remainder = (1)(1)3 + (−2)(1)2 + (−5)(1) + 6 = 1−2−5+6 = 0 Hence (x − 1) is a factor of (x 3 − 2x 2 − 5x + 6) (x 3 − 2x 2

− 5x + 6) is divided by (x + 2), (b) When the remainder is given by (1)(−2)3 + (−2)(−2)2 + (−5)(−2) + 6

Hence (x − 3) is a factor. (iii) Using the remainder theorem, when (x 3 − 2x 2 − 5x + 6) is divided by (x − 3), the remainder is given by ap 3 + bp 2 + cp + d, where a = 1, b = −2, c = −5, d = 6 and p = 3. Hence the remainder is: 1(3)3 + (−2)(3)2 + (−5)(3) + 6 = 27 − 18 − 15 + 6 = 0 Hence (x − 3) is a factor.

= −8 − 8 + 10 + 6 = 0 Thus (x3 − 2x2 − 5x + 6)

Hence (x + 2) is also a factor of: (x 3 − 2x 2 − 5x + 6)

= (x − 1)(x + 2)(x − 3)

Therefore (x − 1)(x + 2)( ) = x 3 − 2x 2 − 5x + 6 To determine the third factor (shown blank) we could (i) divide (x 3 − 2x 2 − 5x + 6) by (x − 1) (x + 2) or (ii) use the factor theorem where f (x) = x 3 − 2x 2 − 5x + 6 and hoping to choose a value of x which makes f (x) = 0 or (iii) use the remainder theorem, again hoping to choose a factor (x − p) which makes the remainder zero (i) Dividing (x 3 − 2x 2 − 5x + 6) by (x 2 + x − 2) gives: x2 + x

−2

x −3 x 3 − 2x 2 − 5x

+6

x 3 + x 2 − 2x ___________ − 3x 2 − 3x + 6 − 3x 2 − 3x + 6 _____________ . . . _____________ Thus (x3 − 2x2 − 5x + 6) = (x − 1)(x + 2)(x − 3) (ii) Using the factor theorem, we let f (x) = x 3 − 2x 2 − 5x + 6 Then

f (3) = 33 − 2(3)2 − 5(3) + 6 = 27 − 18 − 15 + 6 = 0

Now try the following exercise Exercise 27 Further problems on the remainder theorem 1. Find the remainder when 3x 2 − 4x + 2 is divided by: (a) (x − 2) (b) (x + 1) [(a) 6 (b) 9] 2. Determine the remainder when x 3 − 6x 2 + x − 5 is divided by: (a) (x + 2) (b) (x − 3) [(a) −39 (b) −29] 3. Use the remainder theorem to ﬁnd the factors of x 3 − 6x 2 + 11x − 6 [(x − 1)(x − 2)(x − 3)] 4. Determine the factors of x 3 + 7x 2 + 14x + 8 and hence solve the cubic equation: x 3 + 7x 2 + 14x + 8 = 0 [x = −1, x = −2 and x = −4] 5. Determine the value of ‘a’ if (x + 2) is a factor [a = −3] of (x 3 − ax 2 + 7x + 10) 6. Using the remainder theorem, solve the equation: 2x 3 − x 2 − 7x + 6 = 0 [x = 1, x = −2 and x = 1.5]

Chapter 7

Partial fractions 7.1

Introduction to partial fractions

By algebraic addition, 3 (x + 1) + 3(x − 2) 1 + = x −2 x +1 (x − 2)(x + 1) 4x − 5 = 2 x −x −2 The reverse process of moving from

4x − 5 to x2 − x −2

1 3 + is called resolving into partial fractions. x −2 x +1 In order to resolve an algebraic expression into partial fractions: (i) the denominator must factorise (in the above example, x 2 − x − 2 factorises as (x − 2)(x + 1), and (ii) the numerator must be at least one degree less than the denominator (in the above example (4x − 5) is of degree 1 since the highest powered x term is x 1 and (x 2 − x − 2) is of degree 2) When the degree of the numerator is equal to or higher than the degree of the denominator, the numerator

must be divided by the denominator (see Problems 3 and 4). There are basically three types of partial fraction and the form of partial fraction used is summarised in Table 7.1 where f (x) is assumed to be of less degree than the relevant denominator and A, B and C are constants to be determined. (In the latter type in Table 7.1, ax2 + bx + c is a quadratic expression which does not factorise without containing surds or imaginary terms.) Resolving an algebraic expression into partial fractions is used as a preliminary to integrating certain functions (see Chapter 51).

7.2 Worked problems on partial fractions with linear factors Problem 1. Resolve fractions

11 −3x x 2 + 2x − 3

into partial

The denominator factorises as (x − 1)(x + 3) and the numerator is of less degree than the denominator.

Table 7.1 Type

Denominator containing

Expression

Form of partial fraction

1

Linear factors (see Problems 1 to 4)

f (x) (x + a)(x − b)(x + c)

A B C + + (x + a) (x − b) (x + c)

2

Repeated linear factors (see Problems 5 to 7)

f (x) (x + a)3

A C B + + (x + a) (x + a)2 (x + a)3

3

Quadratic factors (see Problems 8 and 9)

DOI: 10.1016/B978-0-08-096562-8.00007-9

f (x) (ax 2 + bx + c)(x

+ d)

Ax + B C + 2 (ax + bx + c) (x + d)

Section 1

56 Engineering Mathematics Thus Let

11 −3x x 2 + 2x − 3

may be resolved into partial fractions.

≡

11 − 3x 11 − 3x A B ≡ ≡ + , 2 x + 2x − 3 (x − 1)(x + 3) (x − 1) (x + 3)

A(x + 3) + B(x − 1) 11 − 3x ≡ (x − 1)(x + 3) (x − 1)(x + 3) by algebraic addition.

Since the denominators are the same on each side of the identity then the numerators are equal to each other. Thus, 11 − 3x ≡ A(x + 3) + B(x − 1)

by algebraic addition Equating the numerators gives: 2x 2 − 9x − 35 ≡A(x − 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x − 2) Let x = −1. Then 2(−1)2 − 9(−1) − 35 ≡ A(−3)(2) + B(0)(2)

To determine constants A and B, values of x are chosen to make the term in A or B equal to zero. When x = 1, then 11 − 3(1) ≡ A(1 +3) +B(0) i.e.

8 = 4A

i.e.

A=2

+ C(0)(−3) −24 = −6A

i.e.

A=

i.e.

2(2)2 − 9(2) − 35 ≡ A(0)(5) + B(3)(5) + C(3)(0)

i.e.

20 =−4B

i.e.

i.e.

B = −5

i.e.

11 − 3x 2 −5 ≡ + x2 + 2x − 3 (x − 1) (x + 3)

2 5 − (x − 1) (x + 3) 2(x + 3) − 5(x − 1) (x − 1)(x + 3) 11 − 3x = 2 x + 2x − 3

=

2x 2 − 9x − 35 into (x + 1)(x − 2)(x + 3) the sum of three partial fractions

Problem 2.

Convert

−45 = 15B B=

−45 = −3 15

Let x = −3. Then 2(−3)2 − 9(−3) − 35 ≡ A(−5)(0) + B(−2)(0)

5 2 − ≡ (x − 1) (x + 3) Check:

−24 =4 −6

Let x = 2. Then

When x = −3, then 11 −3(−3) ≡ A(0) +B(−3 −1)

Thus

B C A + + (x + 1) (x − 2) (x + 3)

A(x − 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x − 2) ≡ (x + 1)(x − 2)(x + 3)

where A and B are constants to be determined, i.e.

2x 2 − 9x − 35 (x + 1)(x − 2)(x + 3)

Let

+ C(−2)(−5) i.e.

10 = 10C

i.e.

C= 1

Thus

2x 2 − 9x − 35 (x + 1)(x − 2)(x + 3) ≡

Problem 3. fractions

Resolve

3 1 4 − + (x + 1) (x − 2) (x + 3) x2 + 1 into partial +2

x 2 − 3x

The denominator is of the same degree as the numerator. Thus dividing out gives: 1 x 2 − 3x + 2 x 2 +1 x 2 − 3x + 2 __________ 3x − 1

Let

For more on polynomial division, see Section 6.1, page 49. x2 + 1 3x − 1 ≡ 1+ 2 2 x − 3x + 2 x − 3x + 2

Hence

≡ 1+ Let

3x − 1 (x − 1)(x − 2)

3x − 1 A B ≡ + (x − 1)(x − 2) (x − 1) (x − 2) A(x − 2) + B(x − 1) ≡ (x − 1)(x − 2)

Let x = 1.

Then

A B x − 10 ≡ + (x + 2)(x − 1) (x + 2) (x − 1) A(x − 1) + B(x + 2) ≡ (x + 2)(x − 1)

Equating the numerators gives: x − 10 ≡ A(x − 1) + B(x + 2) Let x = −2. Then Let x = 1.

Then

Thus

2 = −A

−9 = 3B B = −3

i.e. Hence

−12 = −3A A=4

i.e.

Equating numerators gives: 3x − 1 ≡ A(x − 2) + B(x − 1)

x 3 − 2x 2 − 4x − 4 x − 10 ≡ x −3+ 2 x2 + x − 2 x +x −2 x − 10 ≡ x −3+ (x + 2)(x − 1)

Thus

x − 10 4 3 ≡ − (x + 2)(x − 1) (x + 2) (x − 1) x3 − 2x2 − 4x − 4 x2 + x − 2 3 4 − ≡ x−3+ (x + 2) (x − 1)

A = −2

i.e. Let x = 2.

Then

Now try the following exercise

5=B

Hence

−2 5 3x − 1 ≡ + (x − 1)(x − 2) (x − 1) (x − 2)

Thus

x2 + 1 2 5 ≡1 − + 2 x − 3x + 2 (x − 1) (x − 2)

Problem 4. Express fractions

x 3 − 2x 2 − 4x − 4 in partial x2 + x −2

The numerator is of higher degree than the denominator. Thus dividing out gives: x2 + x − 2

Exercise 28

Resolve the following into partial fractions: 2 2 12 − 1. x2 − 9 (x − 3) (x + 3) 4(x − 4) 5 1 2. − x 2 − 2x − 3 (x + 1) (x − 3) x 2 − 3x + 6 3 2 4 3. + − x(x − 2)(x − 1) x (x − 2) (x − 1) 4.

x −3 x 3 − 2x 2 − 4x − 4 x 3 + x 2 − 2x −3x 2 − 2x − 4 −3x 2 − 3x + 6 x − 10

Further problems on partial fractions with linear factors

5.

3(2x 2 − 8x − 1) (x + 4)(x + 1)(2x − 1) 3 2 7 − − (x + 4) (x + 1) (2x − 1) x 2 + 9x + 8 2 6 1+ + x2 + x −6 (x + 3) (x − 2)

57

Section 1

Partial fractions

Section 1

58 Engineering Mathematics

6. 7.

x 2 − x − 14 x 2 − 2x − 3

2 3 1− + (x − 3) (x + 1)

3x 3 − 2x 2 − 16x + 20 (x − 2)(x + 2) 3x − 2 +

The denominator is a combination of a linear factor and a repeated linear factor.

Let 1 5 − (x − 2) (x + 2)

5x 2 − 2x − 19 (x + 3)(x − 1)2 A B C ≡ + + (x + 3) (x − 1) (x − 1)2 ≡

7.3

Worked problems on partial fractions with repeated linear factors

Problem 5. fractions

2x + 3 Resolve into partial (x − 2)2

2x + 3 A B Let ≡ + (x − 2)2 (x − 2) (x − 2)2 A(x − 2) + B ≡ (x − 2)2 Equating the numerators gives: 2x + 3 ≡ A(x − 2) + B Then 7 = A(0) + B B=7

i.e.

2x + 3 ≡ A(x − 2) + B ≡ Ax − 2A + B Since an identity is true for all values of the unknown, the coefﬁcients of similar terms may be equated. Hence, equating the coefﬁcients of x gives: 2 = A [Also, as a check, equating the constant terms gives: 3 =−2A + B. When A =2 and B =7, RHS =−2(2) +7 = 3 =LHS] 2x + 3

2 7 ≡ Hence + 2 (x − 2) (x − 2) (x − 2)2 5x 2 − 2x − 19 as the sum (x + 3)(x − 1)2 of three partial fractions

Problem 6.

by algebraic addition Equating the numerators gives: 5x 2 − 2x − 19 ≡ A(x − 1)2 + B(x + 3)(x − 1) + C(x + 3)

(1)

Let x = −3. Then 5(−3)2 − 2(−3) − 19 ≡ A(−4)2 + B(0)(−4) + C(0)

The denominator contains a repeated linear factor, (x − 2)2

Let x = 2.

A(x − 1)2 + B(x + 3)(x − 1) + C(x + 3) (x + 3)(x − 1)2

Express

i.e.

32 = 16A

i.e.

A=2

Let x = 1. Then 5(1)2 − 2(1) − 19 ≡ A(0)2 + B(4)(0) + C(4) i.e.

−16 = 4C

i.e.

C = −4

Without expanding the RHS of equation (1) it can be seen that equating the coefﬁcients of x 2 gives: 5 =A + B, and since A = 2, B = 3 [Check: Identity (1) may be expressed as: 5x 2 − 2x − 19 ≡ A(x 2 − 2x + 1) + B(x 2 + 2x − 3) + C(x + 3) i.e. 5x 2 − 2x − 19 ≡ Ax 2 − 2Ax + A + Bx 2 + 2Bx − 3B + Cx + 3C Equating the x term coefﬁcients gives: −2 ≡ −2A + 2B + C When A = 2, B = 3 and C =−4 then −2A +2B +C = −2(2) +2(3) −4 =−2 =LHS Equating the constant term gives: −19 ≡ A − 3B + 3C RHS = 2 − 3(3) + 3(−4) = 2 − 9 − 12 = −19 = LHS]

Partial fractions 5x2 −2x − 19

Now try the following exercise

(x + 3)(x − 1)2 ≡

Exercise 29

2 3 4 + − (x + 3) (x − 1) (x − 1)2

Problem 7. Resolve fractions

3x 2 + 16x + 15 into partial (x + 3)3

+ 15 A C B ≡ + + 3 2 (x + 3) (x + 3) (x + 3) (x + 3)3 A(x + 3)2 + B(x + 3) + C (x + 3)3

2.

x 2 + 7x + 3 x 2 (x + 3)

3.

5x 2 − 30x + 44 (x − 2)3

18 +21x − x 2 (x − 5)(x + 2)2

3(−3)2 + 16(−3) + 15 ≡ A(0)2 + B(0) + C

4.

1 2 1 + − x 2 x (x + 3)

3 4 2 − + (x − 5) (x + 2) (x + 2)2

−6 = C

i.e.

7 4 − (x + 1) (x + 1)2

4 10 5 + − 2 (x − 2) (x − 2) (x − 2)3

(1)

Let x = −3. Then

1.

Equating the numerators gives: 3x 2 + 16x + 15 ≡ A(x + 3)2 + B(x + 3) + C

4x − 3 (x + 1)2

Let

≡

Further problems on partial fractions with repeated linear factors

Resolve the following:

3x 2 + 16x

Section 1

Hence

59

Identity (1) may be expanded as:

3x 2 + 16x + 15 ≡A(x 2 + 6x + 9) + B(x + 3) + C i.e. 3x 2 + 16x + 15 ≡Ax 2 + 6Ax + 9A + Bx + 3B +C Equating the coefﬁcients of x 2 terms gives:

7.4

Worked problems on partial fractions with quadratic factors

3=A Equating the coefﬁcients of x terms gives: 16 = 6A + B

Problem 8. Express fractions

Since A = 3, B = −2 [Check: equating the constant terms gives: 15 = 9A + 3B + C When A = 3, B = −2 and C =−6, 9A + 3B + C = 9(3) + 3(−2) + (−6)

The denominator is a combination of a quadratic factor, (x 2 + 2), which does not factorise without introducing imaginary surd terms, and a linear factor, (x + 1). Let 7x 2 + 5x + 13 Ax + B C ≡ 2 + 2 (x + 2)(x + 1) (x + 2) (x + 1)

= 27 − 6 − 6 = 15 = LHS] Thus

3x2 + 16x + 15 (x + 3)

3

3 6 2 ≡ − − 2 (x + 3) (x + 3) (x + 3)3

7x 2 + 5x + 13 in partial (x 2 + 2)(x + 1)

≡

(Ax + B)(x + 1) + C(x 2 + 2) (x 2 + 2)(x + 1)

Equating numerators gives: 7x 2 + 5x + 13 ≡ (Ax + B)(x + 1) + C(x 2 + 2) (1) Let x = −1. Then 7(−1)2 + 5(−1) + 13 ≡ (Ax + B)(0) + C(1 + 2)

Section 1

60 Engineering Mathematics i.e.

15 = 3C

i.e.

C=5

Since B = 1, D = 3 Equating the coefﬁcients of x terms gives: 6 = 3A

Identity (1) may be expanded as: 7x 2 + 5x + 13 ≡ Ax 2 + Ax + Bx + B + Cx 2 + 2C Equating the coefﬁcients of x 2 terms gives: 7 = A + C, and since C = 5, A = 2

i.e.

A=2

From equation (1), since A = 2, C= −4 Hence

3 + 6x + 4x2 − 2x 3 x2 (x2 + 3)

Equating the coefﬁcients of x terms gives: 5 = A + B, and since A = 2, B = 3 [Check: equating the constant terms gives:

≡

1 2 −4x + 3 + 2+ 2 x x x +3

≡

2 3 − 4x 1 + 2+ 2 x x x +3

13 = B + 2C When B =3 and C =5, B +2C =3 + 10 =13 =LHS] Hence

2x + 3 5 7x2 + 5x + 13 ≡ + (x2 + 2)(x + 1) (x2 + 2) (x + 1)

Problem 9.

Resolve

Exercise 30

Resolve the following:

3 +6x + 4x 2 − 2x 3 into x 2 (x 2 + 3)

x 2 − x − 13 (x 2 + 7)(x − 2)

2.

6x − 5 (x − 4)(x 2 + 3)

3 + 6x + 4x 2 − 2x 3 x 2 (x 2 + 3)

3.

Cx + D B A + 2+ 2 x x (x + 3)

15 + 5x + 5x 2 − 4x 3 x 2 (x 2 + 5)

4.

x 3 + 4x 2 + 20x − 7 (x − 1)2 (x 2 + 8)

Terms such as x 2 may be treated as (x + 0)2, i.e. they are repeated linear factors

≡ ≡

Ax(x 2 + 3) + B(x 2 + 3) + (Cx + D)x 2 x 2 (x 2 + 3)

Equating the numerators gives: 3 + 6x + 4x 2 − 2x 3 ≡ Ax(x 2 + 3) + B(x + 3) + (Cx + D)x 2

2

≡ Ax 3 + 3Ax + Bx 2 + 3B + Cx 3 + Dx 2

2x + 3 1 − (x 2 + 7) (x − 2) 1 2−x + (x − 4) (x 2 + 3) 1 3 2 − 5x + 2+ 2 x x (x + 5)

1 − 2x 2 3 + 2 + 2 (x − 1) (x − 1) (x + 8)

5. When solving the differential equation d 2θ dθ − 6 − 10θ = 20 −e2t by Laplace 2 dt dt transforms, for given boundary conditions, the following expression for L{θ} results: 39 2 s + 42s − 40 2 L{θ} = s(s − 2)(s 2 − 6s + 10)

B=1

Equating the coefﬁcients of x 3 terms gives: −2 = A + C Equating the coefﬁcients of

4s 3 −

Let x = 0. Then 3 = 3B i.e.

Further problems on partial fractions with quadratic factors

1.

partial fractions

Let

Now try the following exercise

x2

terms gives:

4= B+ D

(1)

Show that the expression can be resolved into partial fractions to give: 2 1 5s − 3 L{θ} = − + 2 s 2(s − 2) 2(s − 6s + 10)

Chapter 8

Solving simple equations 8.1 Expressions, equations and identities (3x − 5) is an example of an algebraic expression, whereas 3x − 5 = 1 is an example of an equation (i.e. it contains an ‘equals’ sign). An equation is simply a statement that two quantities are 9 equal. For example, 1 m = 1000 mm or F = C + 32 or 5 y = mx + c. An identity is a relationship that is true for all values of the unknown, whereas an equation is only true for particular values of the unknown. For example, 3x − 5 = 1 is an equation, since it is only true when x = 2, whereas 3x ≡ 8x − 5x is an identity since it is true for all values of x. (Note ‘≡’ means ‘is identical to’). Simple linear equations (or equations of the ﬁrst degree) are those in which an unknown quantity is raised only to the power 1. To ‘solve an equation’ means ‘to ﬁnd the value of the unknown’. Any arithmetic operation may be applied to an equation as long as the equality of the equation is maintained.

Solutions to simple equations should always be checked and this is accomplished by substituting the solution into the original equation. In this case, LHS = 4(5) = 20 = RHS. Problem 2. Solve:

2x =6 5

The LHS is a fraction and this can be removed by multiplying both sides of the equation by 5. 2x = 5(6) Hence, 5 5 Cancelling gives: 2x = 30 Dividing both sides of the equation by 2 gives: 2x 30 = i.e. x = 15 2 2 Problem 3. Solve: a − 5 = 8 Adding 5 to both sides of the equation gives: a−5+5 = 8+5

8.2 Worked problems on simple equations

i.e.

Problem 1. Solve the equation: 4x = 20

The result of the above procedure is to move the ‘−5’ from the LHS of the original equation, across the equals sign, to the RHS, but the sign is changed to +.

20 4x = 4 4 (Note that the same operation has been applied to both the left-hand side (LHS) and the right-hand side (RHS) of the equation so the equality has been maintained). Cancelling gives: x = 5, which is the solution to the equation. Dividing each side of the equation by 4 gives:

DOI: 10.1016/B978-0-08-096562-8.00008-0

a = 13

Problem 4. Solve: x + 3 = 7 Subtracting 3 from both sides of the equation gives: x +3−3 = 7−3 i.e.

x=4

Section 1

62 Engineering Mathematics The result of the above procedure is to move the ‘+3’ from the LHS of the original equation, across the equals sign, to the RHS, but the sign is changed to −. Thus a term can be moved from one side of an equation to the other as long as a change in sign is made. Problem 5.

Solve: 6x + 1 = 2x + 9

In such equations the terms containing x are grouped on one side of the equation and the remaining terms grouped on the other side of the equation. As in Problems 3 and 4, changing from one side of an equation to the other must be accompanied by a change of sign. 6x + 1 = 2x + 9

Thus since

6x − 2x = 9 − 1

then

4x = 8

It is often easier, however, to work with positive values where possible. Problem 7.

Removing the bracket gives: 3x − 6 = 9

x=2

Check: LHS of original equation = 6(2) + 1 = 13 RHS of original equation = 2(2) + 9 = 13 Hence the solution x = 2 is correct. Problem 6.

Solve: 4 − 3p = 2p − 11

3x = 15 3x 15 = 3 3 x=5

i.e.

Check: LHS = 3(5 − 2) = 3(3) = 9 = RHS Hence the solution x = 5 is correct. Problem 8.

4 + 11 = 2p + 3p 15 = 5p 15 5p = 5 5

Hence

3 = p or p = 3

Check: LHS = 4 − 3(3) = 4 − 9 = −5 RHS = 2(3) − 11 = 6 − 11 = −5 Hence the solution p = 3 is correct. If, in this example, the unknown quantities had been grouped initially on the LHS instead of the RHS then:

i.e.

and

Solve:

4(2r − 3) − 2(r − 4) = 3(r − 3) − 1 Removing brackets gives: 8r − 12 − 2r + 8 = 3r − 9 − 1 Rearranging gives: 8r − 2r − 3r = −9 − 1 + 12 − 8 3r = −6

i.e.

In order to keep the p term positive the terms in p are moved to the RHS and the constant terms to the LHS. Hence

3x = 9 + 6

Rearranging gives:

8 4x = 4 4 i.e.

Solve: 3(x − 2) = 9

r=

−6 = −2 3

Check: LHS = 4(−4 − 3) − 2(−2 − 4) = −28 + 12 = −16 RHS = 3(−2 − 3) − 1 = −15 − 1 = −16 Hence the solution r = −2 is correct. Now try the following exercise Exercise 31

Further problems on simple equations

Solve the following equations: 1.

2x + 5 = 7

[1]

−3p − 2p = −11 − 4

2.

8 − 3t = 2

[2]

−5p = −15

3.

−5p −15 = −5 −5

2x − 1 = 5x + 11

4.

7 − 4p = 2p − 3

p = 3, as before

[−4] 2 1 3

5. 2a + 6 − 5a = 0

[2] 1 2

6. 3x − 2 − 5x = 2x − 4 7. 20d − 3 + 3d = 11d + 5 − 8 8. 5( f − 2) − 3(2 f + 5) + 15 = 0

[0]

[6]

11. 2(3g − 5) − 5 = 0 12. 4(3x + 1) = 7(x + 4) − 2(x + 5) 13. 10 + 3(r − 7) = 16 − (r + 2) 14. 8 + 4(x − 1) − 5(x − 3) = 2(5 − 2x)

4 3 = then (3)(5) = 4x, which is a quicker x 5 way of arriving at equation (1) above.)

example, if

Problem 10. Solve:

[−2] 1 2 2

[2] 1 6 4

The LCM of the denominators is 20. Multiplying each term by 20 gives: 2y 3 20 + 20 + 20(5) 5 4 3y 1 − 20 = 20 20 2 Cancelling gives: 4(2y) + 5(3) + 100 = 1 − 10(3y)

[−3] i.e.

8y + 30y = 1 − 15 − 100 38y = −114

3 4 = x 5

y=

The lowest common multiple (LCM) of the denominators, i.e. the lowest algebraic expression that both x and 5 will divide into, is 5x.

Check: LHS =

Multiplying both sides by 5x gives: 3 4 5x = 5x x 5

= RHS =

Cancelling gives:

Check: LHS =

3 4 3 = =3 3 15 15 3 4 4

−6 3 2(−3) 3 + +5 = + +5 5 4 5 4 −9 11 +5 = 4 20 20 3(−3) 1 9 11 1 − = + =4 20 2 20 2 20

(1)

15 4x = 4 4 15 3 x= or 3 4 4

−114 = −3 38

Hence the solution y = −3 is correct.

15 = 4x

i.e.

8y + 15 + 100 = 1 − 30y

Rearranging gives:

8.3 Further worked problems on simple equations Problem 9. Solve:

1 3y 2y 3 + +5 = − 5 4 20 2

[−10]

9. 2x = 4(x − 3) 10. 6(2 − 3y) − 42 = −2(y − 1)

=

63

12 4 = = RHS 15 5

(Note that when there is only one fraction on each side of an equation ‘cross-multiplication’ can be applied. In this

Problem 11. Solve:

3 4 = t − 2 3t + 4

By ‘cross-multiplication’:

3(3t + 4) = 4(t − 2)

Removing brackets gives:

9t + 12 = 4t − 8

Rearranging gives:

9t − 4t = −8 − 12 5t = −20

i.e. t=

−20 = −4 5

Section 1

Solving simple equations

Section 1

64 Engineering Mathematics Check: LHS =

3 1 3 = =− −4 − 2 −6 2

4 4 RHS = = 3(−4) + 4 −12 + 4 4 1 = =− −8 2 Hence the solution t = −4 is correct. Problem 12.

Solve:

√

x =2

√ [ x = 2√is not a ‘simple equation’ since the power of x is 12 i.e x = x (1/2) ; however, it is included here since it occurs often in practise]. Wherever square root signs are involved with the unknown quantity, both sides of the equation must be squared. Hence √ ( x )2 = (2)2 i.e. Problem 13.

x=4 √ Solve: 2 2 = 8

To avoid possible errors it is usually best to arrange the term containing the square root on its own. Thus

i.e.

√ 8 2 d = 2 2 √ d=4

Squaring both sides gives: d = 16, which may be checked in the original equation Problem 14.

Solve: x 2 = 25

This problem involves a square term and thus is not a simple equation (it is, in fact, a quadratic equation). However the solution of such an equation is often required and is therefore included here for completeness. Whenever a square of the unknown is involved, the square root of both sides of the equation is taken. Hence √ x 2 = 25 i.e.

square root of a number is required there are always two answers, one positive, the other negative. The solution of x 2 = 25 is thus written as x = ±5 Problem 15.

15 2 = 2 4t 3

‘Cross-multiplying’ gives:

15(3) = 2(4t 2 ) 45 = 8t 2

i.e.

45 = t2 8 t 2 = 5.625

i.e. Hence t = ﬁgures.

√

5.625 = ±2.372, correct to 4 signiﬁcant

Now try the following exercise Exercise 32

Further problems on simple equations

Solve the following equations: 1.

3 2 5 2+ y =1+ y+ 4 3 6

2.

1 1 (2x − 1) + 3 = 4 2

3.

1 1 2 (2 f − 3) + ( f − 4) + =0 5 6 15

4.

1 1 1 (3m−6) − (5m + 4) + (2m − 9) = −3 3 4 5 [12]

5. 6.

x x − =2 3 5 y y y 1− =3+ − 3 3 6

[−2] −4

1 2

[2]

[15] [−4]

7.

1 7 1 + = 3n 4n 24

8.

x +3 x −3 = +2 4 5

9.

7 5− y y + = 5 20 4

[2]

v−2 1 = 2v − 3 3

[3]

x =5

However, x = −5 is also a solution of the equation because (−5) × (−5) = +25. Therefore, whenever the

Solve:

10.

[2] [13]

11.

2 3 = a − 3 2a + 1

[−11]

x x +6 x +3 − = 4 5 2 √ 13. 3 t = 9 √ 3 x √ = −6 14. 1− x x 15. 10 = 5 −1 2

[−6]

12.

16. 16 =

[9]

0.928 − 0.8 = 32α

[10]

0.128 = 32α

t2 9

[±12]

y+2 1 = y−2 2

18.

11 8 = 5+ 2 2 x

[±4]

8.4 Practical problems involving simple equations Problem 16. A copper wire has a length l of 1.5 km, a resistance R of 5 and a resistivity of 17.2 × 10−6 mm. Find the cross-sectional area, a, of the wire, given that R = ρl/a

Hence

α=

0.128 = 0.004 32

Problem 18. The distance s metres travelled in time t seconds is given by the formula: s = ut + 12 at 2 , where u is the initial velocity in m/s and a is the acceleration in m/s2 . Find the acceleration of the body if it travels 168 m in 6 s, with an initial velocity of 10 m/s 1 s = ut + at 2 , and s = 168, u = 10 and t = 6 2 Hence

1 168 = (10)(6) + a(6)2 2 168 = 60 + 18a

Since R = ρl/a then

168 − 60 = 18a

(17.2 × 10−6 mm)(1500 × 103 mm) a

From the units given, a is measured in

and

0.928 = 0.8[1 + α(40)] 0.928 = 0.8 + (0.8)(α)(40)

17.

Thus

Since Rt = R0 (1 + αt ) then

[4]

1 −3 3

5 =

Problem 17. The temperature coefﬁcient of resistance α may be calculated from the formula Rt = R0 (1 + αt ). Find α given Rt = 0.928, R0 = 0.8 and t = 40

mm2 .

5a = 17.2 × 10−6 × 1500 × 103 a=

108 = 18a a=

108 =6 18

Hence the acceleration of the body is 6 m/s2

17.2 × 10−6 × 1500 × 103

5 17.2 × 1500 × 103 = 106 × 5 17.2 × 15 = 5.16 = 10 × 5

Hence the cross-sectional area of the wire is 5.16 mm2

Problem 19. When three resistors in an electrical circuit are connected in parallel the total resistance RT is given by: 1 1 1 1 = + + RT R1 R2 R3 Find the total resistance when R1 = 5 , R2 = 10 and R3 = 30

65

Section 1

Solving simple equations

66 Engineering Mathematics

Section 1

1 1 1 1 + = + RT 5 10 30 6 + 3 + 1 10 1 = = = 30 30 3

(b) Find the value of R3 given that RT = 3 , R1 = 5 and R2 = 10 . [(a) 1.8 (b) 30 ]

Taking the reciprocal of both sides gives: RT = 3 1 1 1 1 = + + the LCM of the RT 5 10 30 denominators is 30RT Alternatively, if

Hence 30RT

1 RT

1 1 = 30RT + 30RT 5 10 + 30RT

1 30

Cancelling gives: 30 = 6RT + 3RT + RT 30 = 10RT RT =

30 = 3, as above. 10

5. Ohm’s law may be represented by I = V/R, where I is the current in amperes, V is the voltage in volts and R is the resistance in ohms. A soldering iron takes a current of 0.30 A from a 240 V supply. Find the resistance of the element. [800 ]

8.5 Further practical problems involving simple equations Problem 20. The extension x m of an aluminium tie bar of length l m and cross-sectional area A m2 when carrying a load of F newtons is given by the modulus of elasticity E = Fl/ Ax. Find the extension of the tie bar (in mm) if E = 70 × 109 N/m2 , F = 20 × 106 N, A = 0.1 m2 and l = 1.4 m E = Fl/Ax, hence

Now try the following exercise Exercise 33

Practical problems involving simple equations

1. A formula used for calculating resistance of a cable is R = (ρl)/a. Given R = 1.25, l = 2500 and a = 2 × 10−4 ﬁnd the value of ρ. [10−7 ] 2. Force F newtons is given by F = ma, where m is the mass in kilograms and a is the acceleration in metres per second squared. Find the acceleration when a force of 4 kN is applied to a mass of 500 kg. [8 m/s2 ] 3.

PV = mRT is the characteristic gas equation. Find the value of m when P = 100 × 103 , V = 3.00, R = 288 and T = 300. [3.472]

4. When three resistors R1 , R2 and R3 are connected in parallel the total resistance RT is 1 1 1 1 determined from = + + RT R1 R2 R3 (a) Find the total resistance when R1 = 3 , R2 = 6 and R3 = 18 .

70 × 109

N (20 × 106 N)(1.4 m) = m2 (0.1 m2 )(x) (the unit of x is thus metres)

70 × 109 × 0.1 × x = 20 × 106 × 1.4

Cancelling gives:

x=

20 × 106 × 1.4 70 × 109 × 0.1

x=

2 × 1.4 m 7 × 100

=

2 × 1.4 × 1000 mm 7 × 100

Hence the extension of the tie bar, x = 4 mm Problem 21. Power in a d.c. circuit is given by V2 P= where V is the supply voltage and R is the R circuit resistance. Find the supply voltage if the circuit resistance is 1.25 and the power measured is 320 W

Solving simple equations V2 R

320 =

then

V2 1.25

Squaring both sides gives: 4=

(320)(1.25) = V 2

Supply voltage,

f + 1800 f − 1800

4( f − 1800) = f + 1800

V 2 = 400 √ V = 400 = ±20 V

i.e.

Section 1

Since P =

4 f − 7200 = f + 1800 4 f − f = 1800 + 7200

Problem 22. A formula relating initial and ﬁnal states of pressures, P1 and P2 , volumes V1 and V2, and absolute temperatures, T1 and T2 , of an ideal P1 V1 P2 V2 gas is = . Find the value of P2 given T1 T2 3 P1 = 100 × 10 , V1 = 1.0, V2 = 0.266, T1 = 423 and T2 = 293 P2 V2 P1 V1 = T1 T2

Since

then

P2 (0.266) (100 × 103 )(1.0) = 423 293

3 f = 9000 f =

Now try the following exercise

(100 × 10 )(1.0)(293) = P2 (0.266)(423) P2 = Hence P2 = 260 × 103

(100 × 103)(1.0)(293) (0.266)(423)

Given R2 = R1 (1 + αt ), ﬁnd α given [0.004] R1 = 5.0, R2 = 6.03 and t = 51.5

2.

If v 2 = u 2 + 2as, ﬁnd u given v = 24, a = −40 and s = 4.05

Since

then

i.e.

D = d 21.5 = 10.75 2=

3.

or 2.6 × 105

Problem 23. The stress f in a material of a thick D f +p cylinder can be obtained from = d f −p Calculate the stress, given that D = 21.5, d = 10.75 and p = 1800

4. 5.

f + 1800 f − 1800

[30]

The relationship between the temperature on a Fahrenheit scale and that on a Celsius scale 9 is given by F = C + 32. Express 113˚F in 5 degrees Celsius. [45˚C] √ If t = 2π w/Sg, ﬁnd the value of S given w = 1.219, g = 9.81 and t = 0.3132 [50] Applying the principle of moments to a beam results in the following equation: F × 3 = (5 − F) × 7

f +p f −p f + 1800 f − 1800

Practical problems involving simple equations

1.

‘Cross-multiplying’ gives: 3

9000 = 3000 3

Hence stress, f = 3000

Exercise 34

where F is the force in newtons. Determine the value of F. [3.5 N] 6.

67

A rectangular laboratory has a length equal to one and a half times its width and a perimeter of 40 m. Find its length and width. [12 m, 8 m]

Section 1

Revision Test 2 This Revision test covers the material contained in Chapters 5 to 8. The marks for each question are shown in brackets at the end of each question. Evaluate: 3xy2 z 3 − 2yz 1 and z = 2 2. Simplify the following: √ 8a 2 b c3 √ √ (a) (2a)2 b c

1.

when

4 x= , 3

(b) 3x + 4 ÷2x + 5 ×2 − 4x 3.

y =2

7.

(3)

(a) (x − 2) (b) (x + 1) Hence factorise the cubic expression.

(6)

Simplify

9.

Resolve the following into partial fractions: (a)

(a) (2x − y)2 (b) 4ab −[3{2(4a −b) + b(2 −a)}]

(5)

4.

Factorise: 3x 2 y + 9xy2 + 6xy3

(3)

5.

If x is inversely proportional to y and x = 12 when y = 0.4, determine

(c) 10.

(b)

(5)

3−x (x 2 + 3)(x

+ 3)

x 3 − 6x

+9 x2 + x − 2

(24)

Solve the following equations: (b) 4(k − 1) − 2(3k + 2) + 14 =0 s +1 a 2a = 1 (d) =2 (c) − 2 5 s −1

(4)

Factorise x 3 + 4x 2 + x − 6 using the factor theorem. Hence solve the equation (6) x 3 + 4x 2 + x − 6 =0

x − 11 x2 − x − 2

(7)

(a) 3t − 2 =5t + 4

(a) the value of x when y is 3, and

6.

6x 2 + 7x − 5 by dividing out. 2x − 1

8.

Remove the brackets in the following expressions and simplify:

(b) the value of y when x = 2.

Use the remainder theorem to ﬁnd the remainder when 2x 3 + x 2 − 7x − 6 is divided by

11.

(13)

A rectangular football pitch has its length equal to twice its width and a perimeter of 360 m. Find its length and width. (4)

Chapter 9

Solving simultaneous equations 9.1 Introduction to simultaneous equations Only one equation is necessary when ﬁnding the value of a single unknown quantity (as with simple equations in Chapter 8). However, when an equation contains two unknown quantities it has an inﬁnite number of solutions. When two equations are available connecting the same two unknown values then a unique solution is possible. Similarly, for three unknown quantities it is necessary to have three equations in order to solve for a particular value of each of the unknown quantities, and so on. Equations that have to be solved together to ﬁnd the unique values of the unknown quantities, which are true for each of the equations, are called simultaneous equations. Two methods of solving simultaneous equations analytically are: (a) by substitution, and (b) by elimination. (A graphical solution of simultaneous equations is shown in Chapter 31 and determinants and matrices are used to solve simultaneous equations in Chapter 62.)

9.2

Worked problems on simultaneous equations in two unknowns

Problem 1. Solve the following equations for x and y, (a) by substitution, and (b) by elimination: x + 2y = −1 (1) 4x − 3y = 18 (2) DOI: 10.1016/B978-0-08-096562-8.00009-2

(a)

By substitution From equation (1): x = −1 −2y Substituting this expression for x into equation (2) gives: 4(−1 − 2y) − 3y = 18 This is now a simple equation in y. Removing the bracket gives: −4 − 8y − 3y = 18 −11y = 18 + 4 = 22 y=

22 = −2 −11

Substituting y = −2 into equation (1) gives: x + 2(−2) = −1 x − 4 = −1 x = −1 + 4 = 3 Thus x = 3 and y = −2 is the solution to the simultaneous equations. (Check: In equation (2), since x = 3 and y = −2, LHS = 4(3) − 3(−2) = 12 + 6 =18 =RHS.) (b) By elimination x + 2y = −1

(1)

4x − 3y = 18

(2)

If equation (1) is multiplied throughout by 4 the coefﬁcient of x will be the same as in equation (2), giving: 4x + 8y = −4

(3)

70 Engineering Mathematics

Section 1

Subtracting equation (3) from equation (2) gives: 4x − 3y = 18

(2)

4x + 8y = −4 ____________ 0 − 11y = 22 ____________

(3)

22 = −2 −11 (Note, in the above subtraction, 18 − (−4) = 18 + 4 =22.) Substituting y = −2 into either equation (1) or equation (2) will give x = 3 as in method (a). The solution x =3, y = −2 is the only pair of values that satisﬁes both of the original equations. Hence y =

Problem 2. Solve, by a substitution method, the simultaneous equations: 3x − 2y = 12

(1)

x + 3y = −7

(2)

If equation (1) is multiplied throughout by 2 and equation (2) by 3, then the coefﬁcient of x will be the same in the newly formed equations. Thus 2 × equation (1) gives:

6x + 8y = 10

(3)

3 × equation (2) gives:

6x − 15y = −36

(4)

Equation (3) −equation (4) gives: 0 + 23y = 46 y=

i.e.

46 =2 23

(Note +8y − −15y = 8y + 15y = 23y and 10 −(−36) = 10 +36 =46. Alternatively, ‘change the signs of the bottom line and add’.) Substituting y = 2 in equation (1) gives: 3x + 4(2) = 5 from which 3x = 5 − 8 = −3 x = −1

From equation (2), x = −7 −3y

and

Substituting for x in equation (1) gives:

Checking in equation (2), left-hand side = 2(−1) − 5(2) = −2 − 10 =−12 = right-hand side.

3(−7 − 3y) − 2y = 12 i.e.

Hence x = −1 and y = 2 is the solution of the simultaneous equations. The elimination method is the most common method of solving simultaneous equations.

−21 − 9y − 2y = 12 −11y = 12 + 21 = 33

33 = −3 −11 Substituting y = −3 in equation (2) gives:

Hence

y=

Problem 4.

Solve:

x + 3(−3) = −7 i.e. Hence

x − 9 = −7 x = −7 + 9 = 2

Thus x = 2, y= −3 is the solution of the simultaneous equations. (Such solutions should always be checked by substituting values into each of the original two equations.) Problem 3. Use an elimination method to solve the simultaneous equations: 3x + 4y =

5

2x − 5y = −12

(1)

6x + 5y = 29

(2)

When equation (1) is multiplied by 5 and equation (2) by 2 the coefﬁcients of y in each equation are numerically the same, i.e. 10, but are of opposite sign. 5 × equation (1) gives:

35x − 10y = 130

(3)

2 × equation (2) gives:

12x + 10y = 58 _______________ 47x + 0 = 188 _______________

(4)

Adding equation (3) and (4) gives:

(1) (2)

7x − 2y = 26

Hence x =

188 =4 47

(5)

71

[Note that when the signs of common coefﬁcients are different the two equations are added, and when the signs of common coefﬁcients are the same the two equations are subtracted (as in Problems 1 and 3).]

9.3 Further worked problems on simultaneous equations

Substituting x = 4 in equation (1) gives:

Problem 5. Solve:

7(4) − 2y = 26 28 − 2y = 26

3 p = 2q

(1)

4 p + q + 11 = 0

(2)

Rearranging gives:

28 − 26 = 2y

3 p − 2q = 0

2 = 2y Hence

Section 1

Solving simultaneous equations

(3)

4 p + q = −11

y=1

(4)

Multiplying equation (4) by 2 gives: Checking, by substituting x = 4 and y = 1 in equation (2), gives: LHS = 6(4) + 5(1) = 24 + 5 = 29 = RHS Thus the solution is x = 4, y = 1, since these values maintain the equality when substituted in both equations.

8 p + 2q = −22 Adding equations (3) and (5) gives: 11 p + 0 = −22 p=

3(−2) = 2q −6 = 2q

Exercise 35 Further problems on simultaneous equations Solve the following simultaneous equations and verify the results.

2.

4.

Hence the solution is p =−2, q = −3 [x = 1, y = 1]

Problem 6. Solve: x 5 + =y 8 2 y 13 − = 3x 3

[s = 2, t = 3]

[x = 3, y = −2]

5x = 2y 3x + 7y = 41

6.

= 0 = RHS

[a = 5, b = 2]

3x − 2y = 13 2x + 5y = −4

5.

Checking, by substituting p =−2 and q = −3 into equation (2) gives:

3s + 2t = 12 4s − t = 5

[x = 2, y = 5]

5c = 1 − 3d 2d + c + 4 =0

−6 = −3 2

LHS = 4(−2) + (−3) + 11 = −8 − 3 + 11

2x + 5y = 7 x + 3y = 4

3.

q=

a +b=7 a −b=3

−22 = −2 11

Substituting p =−2 into equation (1) gives:

Now try the following exercise

1.

(5)

[c = 2, d = −3]

(1) (2)

Whenever fractions are involved in simultaneous equation it is usual to ﬁrstly remove them. Thus, multiplying equation (1) by 8 gives: x 5 8 +8 = 8y 8 2 i.e.

x + 20 = 8y

(3)

Section 1

72 Engineering Mathematics Rearranging gives:

Multiplying equation (2) by 3 gives: 39 − y = 9x

(4)

Rearranging equation (3) and (4) gives:

(3)

160x + 120y = 108

(4)

Multiplying equation (3) by 2 gives:

x − 8y = −20

(5)

9x + y = 39

(6)

500x − 600y = −150

(5)

Multiplying equation (4) by 5 gives:

Multiplying equation (6) by 8 gives: 72x + 8y = 312

250x − 300y = −75

800x + 600y = 540

(7)

(6)

Adding equations (5) and (6) gives: Adding equations (5) and (7) gives:

1300x + 0 = 390

73x + 0 = 292

x=

292 =4 x= 73

39 3 390 = = = 0.3 1300 130 10

Substituting x = 0.3 into equation (1) gives:

Substituting x = 4 into equation (5) gives:

250(0.3) + 75 − 300y = 0

4 − 8y = −20

75 + 75 = 300y

4 + 20 = 8y

150 = 300y

24 = 8y y=

y=

24 =3 8

Checking x = 0.3, y = 0.5 in equation (2) gives:

Checking: substituting x = 4, y = 3 in the original equations, gives: Equation(1):

Equation(2):

1 4 5 1 LHS = + = + 2 = 3 8 2 2 2 = y = RHS

LHS = 160(0.3) = 48 RHS = 108 − 120(0.5) = 108 − 60 = 48 Hence the solution is x = 0.3, y = 0.5

3 = 13 − 1 = 12 3 RHS = 3x = 3(4) = 12 LHS = 13 −

Now try the following exercise

Exercise 36

Hence the solution is x =4, y = 3 Problem 7.

150 = 0.5 300

Solve:

Further problems on simultaneous equations

Solve the following simultaneous equations and verify the results.

2.5x + 0.75 − 3y = 0

1. 7 p + 11 +2q = 0

1.6x = 1.08 − 1.2y It is often easier to remove decimal fractions. Thus multiplying equations (1) and (2) by 100 gives: 250x + 75 − 300y = 0

(1)

160x = 108 − 120y

(2)

−1 = 3q − 5 p x y 2. + =4 2 3 x y − =0 6 9

[ p =−1, q = −2]

[x = 4, y = 6]

3.

Checking, substituting a = 2 and b = 1 in equation (4) gives:

a − 7 =−2b 2

2 12 =5a + b 3 x 2y 49 + = 4. 5 3 15

5.

[a = 2, b = 3]

3x y 5 − + =0 7 2 7 1.5x − 2.2y = −18

[x = 3, y = 4]

2.4x + 0.6y = 33 6.

[x = 10, y = 15]

LHS = 2 −4(1) = 2 −4 = −2 = RHS Hence a = 2 and b = 1 However, since

1 1 1 = a then x = = x a 2

and since

1 1 1 = b then y = = = 1 y b 1

Hence the solutions is x =

3b − 2.5a = 0.45 1.6a + 0.8b = 0.8

[a = 0.30, b = 0.40]

1 , y = 1, 2

which may be checked in the original equations. Problem 9. Solve: 1 3 + =4 2a 5b

9.4 More difﬁcult worked problems on simultaneous equations

1 4 + = 10.5 a 2b

Problem 8. Solve: Let

2 3 + =7 x y

(1) (2)

Thus equation (1) becomes: 2a + 3b = 7 and equation (2) becomes:

x 3 + y =4 2 5 1 4x + y = 10.5 2

In this type of equation the solutions is easier if a 1 1 substitution is initially made. Let = a and = b x y

a − 4b = −2

(3) (4)

(2)

1 1 = x and =y a b

then

1 4 − = −2 x y

(1)

(3) (4)

To remove fractions, equation (3) is multiplied by 10 giving: x 3 + 10 y = 10(4) 10 2 5 i.e.

5x + 6y = 40

(5)

Multiplying equation (4) by 2 gives: 2a − 8b = −4 Subtracting equation (5) from equation (3) gives: 0 + 11b = 11 i.e.

b=1

Substituting b = 1 in equation (3) gives: 2a + 3 =7 2a = 7 −3 =4 i.e.

a =2

Multiplying equation (4) by 2 gives: (5) 8x + y = 21

(6)

Multiplying equation (6) by 6 gives: 48x + 6y = 126 Subtracting equation (5) from equation (7) gives: 43x + 0 = 86 86 x = =2 43

(7)

Section 1

73

Solving simultaneous equations

Section 1

74 Engineering Mathematics Substituting x = 2 into equation (3) gives:

Now try the following exercise

2 3 + y=4 2 5

Exercise 37

3 y = 4 −1 = 3 5 5 y = (3) = 5 3 1 1 1 Since = x then a = = a x 2 1 1 1 and since = y then b = = b y 5

In Problems 1 to 5, solve the simultaneous equations and verify the results 1.

2.

3. (1)

4.

(3)

Similarly, in equation (2): 33 =4(2x − y) 33 =8x − 4y

i.e.

5.

c+1 d +2 − + 1 =0 4 3 1 − c 3 − d 13 + + =0 5 4 20 3r + 2 2s − 1 11 − = 5 4 5 3 + 2r 5 −s 15 + = 4 3 4

27(1) =4(x + y) 27 =4x + 4y

1 3 + =5 2 p 5q 5 1 35 − = p 2q 2

(2)

To eliminate fractions, both sides of equation (1) are multiplied by 27(x + y) giving: 4 1 = 27(x + y) 27(x + y) x+y 27 i.e.

4 3 − = 18 a b 2 5 + = −4 a b

Solve: 4 1 = x+y 27 1 4 = 2x − y 33

3 2 + = 14 x y

5 3 − = −2 x y

1 1 Hence the solutions is a = , b = 2 5 which may be checked in the original equations. Problem 10.

Further more difﬁcult problems on simultaneous equations

(4)

1 1 x = , y= 2 4

1 1 a = , b=− 3 2

1 1 p= , q= 4 5

[c = 3, d = 4]

1 r = 3, s = 2

3 4 5 6. If 5x − = 1 and x + = ﬁnd the value of y y 2 xy +1 [1] y

Equation (3) + equation (4) gives: 60 =12x, i.e. x =

60 =5 12

Substituting x = 5 in equation (3) gives: 27 = 4(5) + 4y from which 4y = 27 −20 = 7 3 7 and y = = 1 4 4 3 Hence x = 5, y = 1 is the required solution, which may 4 be checked in the original equations.

9.5 Practical problems involving simultaneous equations There are a number of situations in engineering and science where the solution of simultaneous equations is required. Some are demonstrated in the following worked problems. Problem 11. The law connecting friction F and load L for an experiment is of the form F = a L +b, where a and b are constants. When F = 5.6, L =8.0

75

Solving simultaneous equations

Substituting F = 5.6, L =8.0 into F = a L +b gives: 5.6 =8.0a + b

(1)

Substituting F = 4.4, L =2.0 into F = a L +b gives: 4.4 =2.0a + b

−2 = 5 +c

1.2 1 = 6.0 5

1 =10 = LHS 2

1 into equation (1) gives: 5 1 5.6 =8.0 +b 5

Hence the gradient, m = 5 and the y-axis intercept, c = −7

5.6 =1.6 + b

Problem 13. When Kirchhoff’s laws are applied to the electrical circuit shown in Fig. 9.1 the currents I1 and I2 are connected by the equations:

5.6 −1.6 = b i.e.

c = −2 −5 = −7 Checking, substituting m =5 and c = −7 in equation (2), gives: 1 1 (5) + (−7) = 17 − 7 RHS = 3 2 2

1.2 = 6.0a

Substituting a =

1 12 1 1 12 = 2 m from which, m = 2 = 5 1 2 2 2 2 Substituting m = 5 into equation (1) gives:

(2)

Subtracting equation (2) from equation (1) gives:

a=

Subtracting equation (1) from equation (2) gives:

b=4

27 = 1.5I1 + 8(I1 − I2 )

1 Checking, substituting a = and b = 4 in equation (2), 5 gives: 1 RHS = 2.0 + 4 =0.4 + 4 = 4.4 = LHS 5

l1

(2)

l2 (l1 ⫺ l 2)

26 V

8⍀

1 When L = 6.5, F = al + b = (6.5) + 4 5 = 1.3 +4, i.e. F =5.3

2⍀

1.5 ⍀

Problem 12. The equation of a straight line, of gradient m and intercept on the y-axis c, is y = mx + c. If a straight line passes through the point where x = 1 and y = −2, and also through the point where x = 3 12 and y = 10 12 , ﬁnd the values of the gradient and the y-axis intercept Substituting x = 1 and y = −2 into y = mx + c gives: (1)

1 1 Substituting x = 3 and y = 10 into y = mx + c gives: 2 2 1 1 10 = 3 m + c 2 2

(1)

−26 = 2I2 − 8(I1 − I2 )

27 V

1 and b = 4 Hence a = 5

−2 = m + c

Section 1

and when F = 4.4, L =2.0. Find the values of a and b and the value of F when L =6.5

Figure 9.1

Solve the equations to ﬁnd the values of currents I1 and I2 Removing the brackets from equation (1) gives: 27 =1.5I1 + 8I1−8I2 Rearranging gives: 9.5I1 − 8I2 = 27 Removing the brackets from equation (2) gives:

(2)

−26 =2I2 − 8I1 + 8I2

(3)

Section 1

76 Engineering Mathematics Multiplying equation (1) by 2 gives:

Rearranging gives: −8I1 + 10I2 = −26

(4)

Multiplying equation (3) by 5 gives: 47.5I1 − 40I2 = 135

(5)

60 =0 + 4a a=

(6)

Adding equations (5) and (6) gives: 15.5I1 + 0 = 31 I1 =

(3)

Subtracting equation (3) from equation (2) gives:

Multiplying equation (4) by 4 gives: −32I1 + 40I2 = −104

84 =4u +4a

31 =2 15.5

Substituting I1 = 2 into equation (3) gives: 9.5(2) − 8I2 = 27 19 −8I2 = 27 19 − 27 =8I2 −8 =8I2 I 2 = −1 Hence the solution is I 1 = 2 and I 2 = −1 (which may be checked in the original equations).

60 = 15 15

Substituting a = 15 into equation (1) gives: 42 =2u + 2(15) 42 −30 =2u u=

12 =6 2

Substituting a = 15, u =6 in equation (2) gives: RHS =4(6) + 8(15) = 24 +120 =144 =LHS Hence the initial velocity, u = 6 m/s and the acceleration, a =15 m/s2. 1 Distance travelled after 3 s is given by s = ut + at 2 2 where t = 3, u =6 and a = 15 1 Hence s = (6)(3) + (15)(3)2 = 18 +67.5 2 i.e. distance travelled after 3 s = 85.5 m.

Problem 14. The distance s metres from a ﬁxed point of a vehicle travelling in a straight line with constant acceleration, a m/s2 , is given by s = ut + 12 at 2 , where u is the initial velocity in m/s and t the time in seconds. Determine the initial velocity and the acceleration given that s = 42 m when t = 2 s and s = 144 m when t = 4 s. Find also the distance travelled after 3 s 1 Substituting s = 42, t = 2 into s = ut + at 2 gives: 2 1 42 =2u + a(2)2 2 i.e. 42 = 2u +2a (1) 1 Substituting s = 144, t = 4 into s = ut + at 2 gives: 2 1 144 =4u + a(4)2 2 i.e. 144 =4u + 8a (2)

Problem 15. The resistance R of a length of wire at t ◦ C is given by R = R0 (1 + αt ), where R0 is the resistance at 0◦ C and α is the temperature coefﬁcient of resistance in /◦ C. Find the values of α and R0 if R = 30 at 50◦C and R = 35 at 100◦ C Substituting R = 30, t = 50 into R = R0 (1 + αt ) gives: 30 = R0 (1 + 50α)

(1)

Substituting R = 35, t = 100 into R = R0 (1 + αt ) gives: 35 = R0 (1 +100 α)

(2)

Although these equations may be solved by the conventional substitution method, an easier way is to eliminate R0 by division. Thus, dividing equation (1) by equation (2) gives: 30 R0 (1 + 50α) 1 + 50α = = 35 R0 (1 + 100α) 1 +100α

a =52 − 40 = 12

‘Cross-multiplying’ gives:

Hence a =12 and

30(1 + 100α) = 35(1 + 50α)

b =0.4

30 + 3000α = 35 + 1750α 3000α − 1750α = 35 − 30

Now try the following exercise

1250α = 5 α=

i.e.

Exercise 38

5 1 = or 0.004 1250 250

1 into equation (1) gives: 250 1 30 = R0 1 + (50) 250

Substituting α =

1.

In a system of pulleys, the effort P required to raise a load W is given by P = aW + b, where a and b are constants If W = 40 when P = 12 and W = 90 when P = 22, ﬁnd the values of a and b 1 a = , b=4 5

2.

Applying Kirchhoff’s laws to an electrical circuit produces the following equations:

30 = R0 (1.2) R0 =

30 = 25 1.2

1 Checking, substituting α = and R0 = 25 in equa250 tion (2) gives: 1 RHS = 25 1 + (100) 250 = 25(1.4) = 35 = LHS

5 = 0.2I1 + 2(I1 − I2 ) 12 = 3I2 + 0.4I2 − 2(I1 − I2 ) Determine the values of currents I1 and I2 [I1 = 6.47, I2 = 4.62] 3.

Velocity v is given by the formula v = u + at . If v = 20 when t = 2 and v = 40 when t = 7, ﬁnd the values of u and a. Hence ﬁnd the velocity when t = 3.5 [u = 12, a = 4, v = 26]

4.

y = mx + c is the equation of a straight line of slope m and y-axis intercept c. If the line passes through the point where x = 2 and y = 2, and also through the point where x = 5 and y = 12 , ﬁnd the slope and y-axis intercept of 1 the straight line m =− , c=3 2

5.

The resistance R ohms of copper wire at t ◦ C is given by R = R0 (1 + αt ), where R0 is the resistance at 0◦ C and α is the temperature coefﬁcient of resistance. If R = 25.44 at 30◦ C and R = 32.17 at 100◦C, ﬁnd α [α = 0.00426, R0 = 22.56 ] and R0

6.

The molar heat capacity of a solid compound is given by the equation c = a + bT . When c = 52, T = 100 and when c = 172, T = 400. Find the values of a and b [a = 12, b = 0.40]

Thus the solution is α = 0.004/◦ C and R0 =25 . Problem 16. The molar heat capacity of a solid compound is given by the equation c = a + bT, where a and b are constants. When c = 52, T = 100 and when c = 172, T = 400. Determine the values of a and b When c = 52, T = 100, hence 52 = a + 100b

(1)

When c = 172, T = 400, hence 172 = a + 400b

(2)

Equation (2) – equation (1) gives: 120 =300b from which, b =

120 = 0.4 300

Substituting b = 0.4 in equation (1) gives: 52 =a + 100(0.4)

Further practical problems involving simultaneous equations

77

Section 1

Solving simultaneous equations

Chapter 10

Transposition of formulae 10.1 Introduction to transposition of formulae

Rearranging gives: w − x + y = a + b and − x = a + b − w − y Multiplying both sides by −1 gives:

When a symbol other than the subject is required to be calculated it is usual to rearrange the formula to make a new subject. This rearranging process is called transposing the formula or transposition. The rules used for transposition of formulae are the same as those used for the solution of simple equations (see Chapter 8)—basically, that the equality of an equation must be maintained.

10.2 Worked problems on transposition of formulae Problem 1. subject

Transpose p =q +r + s to make r the

(−1)(−x) = (−1)(a + b − w − y) i.e.

x = −a − b + w + y

The result of multiplying each side of the equation by −1 is to change all the signs in the equation. It is conventional to express answers with positive quantities ﬁrst. Hence rather than x = −a − b + w + y, x =w +y − a − b, since the order of terms connected by + and − signs is immaterial. Problem 3. subject

Transpose v = f λ to make λ the fλ=v

Rearranging gives: Dividing both sides by f gives:

The aim is to obtain r on its own on the left-hand side (LHS) of the equation. Changing the equation around so that r is on the LHS gives: q +r +s = p (1) Substracting (q + s) from both sides of the equation gives: q + r + s − (q + s) = p − (q + s) Thus

q +r +s −q −s = p −q −s

i.e. r = p−q−s (2) It is shown with simple equations, that a quantity can be moved from one side of an equation to the other with an appropriate change of sign. Thus equation (2) follows immediately from equation (1) above. Problem 2. subject

If a + b = w − x + y, express x as the

DOI: 10.1016/B978-0-08-096562-8.00010-9

i.e.

fλ v = f f v λ= f

Problem 4. When a body falls freely through a height h, the velocity v is given by v 2 = 2gh. Express this formula with h as the subject Rearranging gives:

2gh = v 2

Dividing both sides by 2g gives:

2gh v2 = 2g 2g h=

i.e.

Problem 5. subject

If I =

v2 2g

V , rearrange to make V the R

V =I R Multiplying both sides by R gives: V R = R(I ) R

Now try the following exercise

Rearranging gives:

Hence

Exercise 39

Further problems on transposition of formulae

Make the symbol indicated the subject of each of the formulae shown and express each in its simplest form.

V = IR F for m m

1.

a +b=c−d −e

(d)

2.

x + 3y = t

( y)

3.

c = 2πr

(r)

4.

y = mx + c

(x)

5.

I = PRT

(T )

6.

I=

E R

(R)

ρl to Problem 7. Rearrange the formula: R = a make (i) a the subject, and (ii) l the subject

7.

S=

a 1 −r

(r)

ρl (i) Rearranging gives: =R a

8.

9 F = C + 32 5

(C)

Problem 6. Transpose: a = Rearranging gives:

F =a m

Multiplying both sides by m gives: F = m(a) i.e. m m

F = ma

Rearranging gives:

ma = F

Dividing both sides by a gives:

ma F = a a F m= a

i.e.

Multiplying both sides by a gives: ρl a = a(R) i.e. ρl = aR a

10.3 Further worked problems on transposition of formulae

Rearranging gives: aR = ρl Dividing both sides by R gives:

i.e.

Dividing both sides by ρ gives:

ft ft = v and =v−u m m Multiplying each side by m gives: ft = m(v − u) i.e. ft = m(v − u) m m Rearranging gives: u +

ρl = R by a gives: a

ρl = aR

i.e.

ft Problem 8. Transpose the formula: v = u + to m make f the subject

aR ρl = R R ρl a= R

(ii) Multiplying both sides of

[d = c − a − b] 1 y = (t − x) 3 c r= 2π y −c x= m I T= PR E R= I ⎤ ⎡ S −a R = ⎢ S ⎥ ⎣ a ⎦ or 1 − S 5 C = (F − 32) 9

aR ρl = ρ ρ l=

aR ρ

Dividing both sides by t gives: ft m = (v − u) t t

i.e.

f=

m (v − u) t

79

Section 1

Transposition of formulae

Section 1

80 Engineering Mathematics Problem 9. The ﬁnal length, l2 of a piece of wire heated through θ ◦ C is given by the formula l2 = l1 (1 + αθ). Make the coefﬁcient of expansion, α, the subject l1 (1 + αθ) = l2

Rearranging gives:

l1 + l1 αθ = l2

Removing the bracket gives:

l1 αθ = l2 − l1

Rearranging gives:

Taking the square root of both sides gives: √ 2k 2 v = m 2k i.e. v= m Problem 12. In a right-angled triangle having sides x, y and hypotenuse z, Pythagoras’ theorem states z 2 = x 2 + y 2 . Transpose the formula to ﬁnd x

Dividing both sides by l1 θ gives: l1 αθ l2 − l1 l 2 − l1 = i.e. α = l1 θ l1 θ l 1θ Problem 10.

A formula for the distance moved 1 by a body is given by: s = (v + u)t . Rearrange the 2 formula to make u the subject

Rearranging gives: Multiplying both sides by 2 gives:

1 (v + u)t = s 2 (v + u)t = 2s

Dividing both sides by t gives: (v + u)t 2s = t t v +u =

i.e. Hence

u=

2s −v t

or

u=

2s t 2s − vt t

Problem 11. A formula for kinetic energy is 1 k = mv 2 . Transpose the formula to make v the 2 subject 1 Rearranging gives: mv 2 = k 2 Whenever the prospective new subject is a squared term, that term is isolated on the LHS, and then the square root of both sides of the equation is taken. Multiplying both sides by 2 gives:

mv 2 = 2k

Dividing both sides by m gives:

mv 2 2k = m m

i.e.

v2 =

2k m

Rearranging gives:

x 2 + y2 = z2

and

x 2 = z2 − y2

Taking the square root of both sides gives: x = z 2 − y2 Problem 13.

Given t = 2π

t , l and π

l ﬁnd g in terms of g

Whenever the prospective new subject is within a square root sign, it is best to isolate that term on the LHS and then to square both sides of the equation. l Rearranging gives: 2π =t g t l = Dividing both sides by 2π gives: g 2π l Squaring both sides gives: = g

t 2π

2 =

t2 4π 2

Cross-multiplying, i.e. multiplying each term by 4π 2 g, gives: 4π 2l = gt 2 or

gt 2 = 4π 2 l

Dividing both sides by t 2 gives:

gt 2 4π 2 l = 2 t2 t

i.e.

g=

4π 2 l t2

Problem 14. √The impedance of an a.c. circuit is given by Z = R 2 + X 2 . Make the reactance, X, the subject

Rearranging gives:

R2 + X 2 = Z R2 + X 2 = Z 2

Squaring both sides gives:

5.

(R2 )

6.

I=

E −e R+ r

(R)

X 2 = Z 2 − R2

Rearranging gives:

Taking the square root of both sides gives: X = Z2 − R2 Problem 15. The volume V of a hemisphere is 2 given by V = πr 3 . Find r in terms of V 3 Rearranging gives: Multiplying both sides by 3 gives:

2 3 πr = V 3 2πr 3 = 3V

Dividing both sides by 2π gives: 2πr 3 2π

=

3V 2π

R=

E −e− Ir I

7.

y = 4ab 2 c2

(b)

8.

a 2 b2 + =1 x 2 y2

(x)

9. t = 2π

l g

10. v 2 = u 2 + 2as r3 =

i.e.

3V 2π

Taking the cube root of both sides gives: √ 3 3 3V 3 3V 3 r = i.e. r = 2π 2π

1 1 1 + = R R1 R2

11.

πR 2 θ A= 360

12.

13.

N= Z=

a+x y

R R1 R2 = R1 − R

E −e −r I y b= 4ac2 ay x= y2 − b2 t 2g l= 2 4π √ u = v 2 − 2as

or R =

(l) (u)

360 A πθ

(R)

R=

(a)

[a = N 2 y − x]

√ L=

R 2 + (2πfL)2 (L)

Now try the following exercise

Z 2 − R2 2π f

Exercise 40 Further problems on transposition of formulae Make the symbol indicated the subject of each of the formulae shown and express each in its simplest form. λ(x − d) (x) 1. y = d d yd x = (y + λ) or x = d + λ λ 2.

3(F − f ) A= L

(f)

AL or f = F − 3 Ml 2 (E) E= 8y I R − R0 (t ) t= R0 α

3F − AL f= 3

3. 4.

Ml 2 y= 8EI R = R0 (1 + αt )

10.4 Harder worked problems on transposition of formulae Problem 16. Transpose the formula a2 x + a2 y to make a the subject p= r Rearranging gives:

a2x + a2 y =p r

Multiplying both sides by r gives:

a 2 x + a 2 y = rp

Factorising the LHS gives:

a 2 (x + y) = rp

Dividing both sides by (x + y) gives: a 2 (x + y) rp rp = i.e. a 2 = (x + y) (x + y) (x + y)

81

Section 1

Transposition of formulae

Section 1

82 Engineering Mathematics Taking the square root of both sides gives: rp a= x+y Problem 17.

Make b the subject of the formula x−y a= √ bd + be

x−y =a Rearranging gives: √ bd + be √ Multiplying both sides by bd + be gives: √ x − y = a bd + be √ or a bd + be = x − y Dividing both sides by a gives:

Squaring both sides gives: bd + be =

x−y a

Dividing both sides by (1 −a) gives: b=

Problem 19.

Transpose the formula V =

make r the subject

b(d + e) =

x−y a

2

Er =V R+r Multiplying both sides by (R + r) gives: Er = V(R + r) Removing the bracket gives: Er = VR + Vr

Er − Vr = VR Factorising gives: r(E − V ) = VR Dividing both sides by (E − V ) gives: r=

2

Dividing both sides by (d + e) gives:

i.e.

b=

2

x−y a (d + e)

b=

(x − y)2 a2 (d + e)

Problem 18.

If a =

the formula

Er to R+r

Rearranging gives:

Factorising the LHS gives:

a 1−a

Rearranging to obtain terms in r on the LHS gives:

√ x−y bd + be = a

Factorising the LHS gives: b(1 − a) =a

Problem 20.

VR E −V

Given that:

D = d

in terms of D, d and f

f +p express p f −p

f+p D = f−p d f+p D2 Squaring both sides gives: = 2 f−p d Rearranging gives:

Cross-multiplying, i.e. multiplying each term by d 2 ( f − p), gives: b make b the subject of 1 +b

b =a 1+b Multiplying both sides by (1 +b) gives:

Rearranging gives:

b = a(1 + b) Removing the bracket gives: b = a + ab Rearranging to obtain terms in b on the LHS gives: b − ab = a

d 2 ( f + p) = D 2 ( f − p) Removing brackets gives: d 2 f + d 2 p = D 2 f − D 2 p Rearranging, to obtain terms in p on the LHS gives: d 2p + D2 p = D2 f − d 2 f Factorising gives: p(d 2 + D 2 ) = f(D 2 − d 2 ) Dividing both sides by (d 2 + D 2 ) gives: p=

f (D2 − d 2 ) (d2 + D2 )

Now try the following exercise Exercise 41 Further problems on transposition of formulae Make the symbol indicated the subject of each of the formulae shown in Problems 1 to 7, and express each in its simplest form. a2m − a2n xy 1. y = (a) a= x m −n 4 M 4 4 4 2. M = π(R −r ) (R) R= +r π 3.

x +y=

4. m = 5. a 2 = 6.

7.

r 3 +r

μL L +rCR b2 − c2 b2

x 1 +r 2 = y 1 −r 2 a + 2b p = q a − 2b

3(x + y) r= (1 − x − y) mrC R (L) L= μ−m c (b) b= √ 1 − a2 x−y (r) r= x+y a( p2 − q 2 ) (b) b = 2( p2 + q 2 )

( r)

8. A formula for the focal length, f , of a convex 1 1 1 lens is = + . Transpose the formula to f u v make v the subject and evaluate v when f = 5 and u =6. uf , 30 v= u− f 9. The quantity of heat, Q, is given by the formula Q = mc(t2 − t1 ). Make t2 the subject of the formula and evaluate t2 when m = 10, t1 = 15, c = 4 and Q = 1600. Q , 55 t2 = t1 + mc 10. The velocity, v, of water in a pipe appears in 0.03Lv 2 the formula h = . Express v as the 2dg

subject of the formula and evaluate v when h = 0.712, L = 150, d = 0.30 and g = 9.81 2dgh v= , 0.965 0.03L 11. The sag S at the centre of a wire is given 3d(l − d) by the formula: S = . Make l the 8 subject of the formula and evaluate l when d = 1.75 and S = 0.80 8S 2 + d, 2.725 l= 3d 12. In an electrical alternating current circuit the impedance Z is given by: Z=

R2

1 2 + ωL − ωC

Transpose the formula to make C the subject and hence evaluate C when Z = 130, R = 120, ω = 314 and L =0.32 ⎤ ⎡ 1 ⎣C = , 63.1×10−6 ⎦ √ 2 2 ω ωL − Z − R 13. An approximate relationship between the number of teeth, T , on a milling cutter, the diameter of cutter, D, and the depth of cut, 12.5 D d, is given by: T = . Determine the D + 4d value of D when T = 10 and d = 4 mm. [64 mm] 14. Make λ, the wavelength of X-rays, the subject of the following formula: √ μ CZ 4 λ5 n = ρ a ⎡ ⎤ 2 aμ ⎣λ = 5 ⎦ ρCZ 4 n

83

Section 1

Transposition of formulae

Chapter 11

Solving quadratic equations Problem 1. Solve the equations: (b) 3x 2 − 11x − 4 =0 by (a) x 2 + 2x − 8 =0 factorisation

11.1 Introduction to quadratic equations As stated in Chapter 8, an equation is a statement that two quantities are equal and to ‘solve an equation’ means ‘to ﬁnd the value of the unknown’. The value of the unknown is called the root of the equation. A quadratic equation is one in which the highest power of the unknown quantity is 2. For example, x 2 − 3x + 1 =0 is a quadratic equation. There are four methods of solving quadratic equations. These are: (i) by factorisation (where possible) (ii) by ‘completing the square’ (iii) by using the ‘quadratic formula’ or (iv) graphically (see Chapter 31).

11.2 Solution of quadratic equations by factorisation Multiplying out (2x + 1)(x − 3) gives 2x 2 − 6x + x − 3, i.e. 2x 2 − 5x − 3. The reverse process of moving from 2x 2 − 5x − 3 to (2x + 1)(x − 3) is called factorising. If the quadratic expression can be factorised this provides the simplest method of solving a quadratic equation. For example, if

2x 2 − 5x − 3 = 0, then,

by factorising:

(2x + 1)(x − 3) = 0

Hence either or

(2x + 1) = 0

i.e.

x =−

(x − 3) = 0

i.e.

x =3

1 2

The technique of factorising is often one of ‘trial and error’. DOI: 10.1016/B978-0-08-096562-8.00011-0

(a)

x 2 + 2x − 8 = 0. The factors of x 2 are x and x. These are placed in brackets thus: (x )(x ) The factors of −8 are +8 and −1, or −8 and +1, or +4 and −2, or −4 and +2. The only combination to given a middle term of +2x is +4 and −2, i.e. x 2 + 2x − 8 = (x + 4)(x − 2) (Note that the product of the two inner terms added to the product of the two outer terms must equal to the middle term, +2x in this case.) The quadratic equation x 2 + 2x − 8 =0 thus becomes (x + 4)(x − 2) = 0. Since the only way that this can be true is for either the ﬁrst or the second, or both factors to be zero, then either

(x + 4) = 0

i.e. x = −4

or

(x − 2) = 0

i.e. x = 2

Hence the roots of x2 + 2x − 8 =0 are x = −4 and 2 (b) 3x 2 − 11x − 4 = 0 The factors of 3x 2 are 3x and x. These are placed in brackets thus: (3x )(x ) The factors of −4 are −4 and +1, or +4 and −1, or −2 and 2. Remembering that the product of the two inner terms added to the product of the two outer terms must equal −11x, the only combination to give this is +1 and −4, i.e. 3x 2 − 11x − 4 = (3x + 1)(x − 4)

The quadratic equation 3x 2 − 11x − 4 = 0 thus becomes (3x + 1)(x − 4) = 0. Hence, either or

(3x + 1) = 0

i.e.

x=−

(x − 4) = 0

i.e.

x=4

(b)

1 3

Hence (5x + 4)(3x − 2) = 0 from which

Problem 2. Determine the roots of: (a) x 2 − 6x + 9 =0, and (b) 4x 2 − 25 =0, by factorisation x 2 − 6x + 9 = 0. Hence (x − 3) (x − 3) = 0, i.e. (x − 3)2 = 0 (the left-hand side is known as a perfect square). Hence x = 3 is the only root of the equation x 2 − 6x + 9 =0.

(b)

4x 2 − 25 =0 (the left-hand side is the difference of two squares, (2x)2 and (5)2 ). Thus (2x + 5)(2x − 5) = 0. Hence either or

(2x + 5) = 0

5 i.e. x = − 2

(2x − 5) = 0

5 i.e. x = 2

15x 2 + 2x − 8 =0. The factors of 15x 2 are 15x and x or 5x and 3x. The factors of −8 are −4 and +2, or 4 and −2, or −8 and +1, or 8 and −1. By trial and error the only combination that works is: 15x 2 + 2x − 8 = (5x + 4)(3x − 2)

and both solutions may be checked in the original equation.

(a)

either

5x + 4 = 0

or

3x − 2 = 0

2 4 or x = 5 3 which may be checked in the original equation.

Hence x = −

1 Problem 4. The roots of quadratic equation are 3 and −2. Determine the equation If the roots of a quadratic equation are α and β then (x − α)(x − β) = 0. Hence if α =

4x 2 + 8x + 3 =0. The factors of 4x 2 are 4x and x or 2x and 2x. The factors of 3 are 3 and 1, or −3 and −1. Remembering that the product of the inner terms added to the product of the two outer terms must equal +8x, the only combination that is true (by trial and error) is:

1 and β = −2, then 3

Problem 3. Solve the following quadratic equations by factorising: (a) 4x 2 + 8x + 3 =0 (b) 15x 2 + 2x − 8 =0. (a)

85

1 (x − (−2)) = 0 x− 3 1 (x + 2) = 0 x− 3 1 2 x 2 − x + 2x − = 0 3 3 5 2 x2 + x − = 0 3 3

Hence

3x2 + 5x − 2 = 0

(4x 2 + 8x + 3) = (2x + 3)(2x + 1) Hence (2x + 3)(2x + 1) = 0 from which, either (2x + 3) = 0

or

(2x + 1) = 0

Problem 5. Find the equations of x whose roots are: (a) 5 and −5 (b) 1.2 and −0.4 (a)

3 2

Thus,

2x = −3, from which, x = −

or

1 2x = −1, from which, x = − 2

which may be checked in the original equation.

If 5 and −5 are the roots of a quadratic equation then: (x − 5)(x + 5) = 0 i.e.

x 2 − 5x + 5x − 25 = 0

i.e.

x2 − 25 = 0

Section 1

Solving quadratic equations

Section 1

86 Engineering Mathematics (b) If 1.2 and −0.4 are the roots of a quadratic equation then: (x − 1.2)(x + 0.4) = 0

An expression such as x 2 or (x + 2)2 or (x − 3)2 is called a perfect square. √ If x 2 = 3 then x = ± 3 √ √ If (x + 2)2 = 5 then x + 2 = ± 5 and x = −2 ± 5 √ √ If (x − 3)2 = 8 then x − 3 = ± 8 and x = 3 ± 8

i.e. x − 1.2x + 0.4x − 0.48 = 0 2

x2 − 0.8x − 0.48 = 0

i.e.

Now try the following exercise Exercise 42

Further problems on solving quadratic equations by factorisation

In Problems 1 to 10, solve the given equations by factorisation. 1.

x 2 + 4x − 32 =0

[4, −8]

2.

x 2 − 16 =0

[4, −4]

3.

(x + 2)2 = 16

4.

2x 2 − x − 3 =0

5.

6x 2 − 5x + 1 = 0

6.

10x 2 + 3x − 4 = 0

7.

x 2 − 4x + 4 =0

8.

21x 2 − 25x = 4

9.

6x 2 − 5x − 4 = 0

10. 8x 2 + 2x − 15 =0

[2, −6] 1 −1, 1 2 1 1 , 2 3 1 4 ,− 2 5 [2] 1 1 1 ,− 3 7 1 4 ,− 3 2 3 5 ,− 4 2

In Problems 11 to 16, determine the quadratic equations in x whose roots are: 11. 3 and 1 12. 2 and −5

(x + a)2 = x 2 + 2ax + a 2 Thus in order to make the quadratic expression x 2 + 2ax into a perfect square it is necessary to add (half the 2 2a 2 or a 2 coefﬁcient of x) i.e. 2 For example, x 2 + 3x becomes a perfect square by 2 3 , i.e. adding 2 2 3 3 2 x 2 + 3x + = x+ 2 2 The method is demonstrated in the following worked problems. Problem 6. the square’

1 1 14. 2 and − 2 2

[4x 2 − 8x − 5 = 0] [x 2 − 36 =0] [x 2 − 1.7x − 1.68 = 0]

Solve 2x 2 + 5x = 3 by ‘completing

The procedure is as follows: 1.

Rearrange the equations so that all terms are on the same side of the equals sign (and the coefﬁcient of the x 2 term is positive). Hence 2x 2 + 5x − 3 =0

[x 2 + 3x − 10 = 0] [x 2 + 5x + 4 = 0]

16. 2.4 and −0.7

Hence if a quadratic equation can be rearranged so that one side of the equation is a perfect square and the other side of the equation is a number, then the solution of the equation is readily obtained by taking the square roots of each side as in the above examples. The process of rearranging one side of a quadratic equation into a perfect square before solving is called ‘completing the square’.

[x 2 − 4x + 3 = 0]

13. −1 and −4

15. 6 and −6

11.3 Solution of quadratic equations by ‘completing the square’

2.

Make the coefﬁcient of the x 2 term unity. In this case this is achieved by dividing throughout by 2. Hence 2x 2 5x 3 + − =0 2 2 2 5 3 i.e. x 2 + x − = 0 2 2

3.

Rearrange the equations so that the x 2 and x terms are on one side of the equals sign and the constant is on the other side, Hence 5 3 x2 + x = 2 2

4.

Add to both sides of the equation (half the coefﬁ5 cient of x)2 . In this case the coefﬁcient of x is . 2 2 5 . Half the coefﬁcient squared is therefore 4 2 2 5 3 5 5 Thus, x 2 + x + = + 2 4 2 4 The LHS is now a perfect square, i.e. x+

5.

=

2 3 5 + 2 4

5 x+ 4

2 =

3 25 24 + 25 49 + = = 2 16 16 16

Taking the square root of both sides of the equation (remembering that the square root of a number gives a ± answer). Thus 5 2 49 = x+ 4 16 i.e.

7.

2

Evaluate the RHS. Thus

6.

5 4

x+

7 5 =± 4 4

Making the coefﬁcient of x 2 unity gives: 9 x2 + x + 4 = 0 2 9 and rearranging gives: x 2 + x = −4 2 Adding to both sides (half the coefﬁcient of x)2 gives: 2 2 9 9 9 2 x + x+ = −4 2 4 4 The LHS is now a perfect square, thus: 17 9 2 81 = −4 = x+ 4 16 16 Taking the square root of both sides gives: 9 17 = ±1.031 x+ = 4 16 9 Hence x = − ± 1.031 4 i.e. x = −1.22 or −3.28, correct to 3 signiﬁcant ﬁgures. Problem 8. By ‘completing the square’, solve the quadratic equation 4.6y 2 + 3.5y − 1.75 =0, correct to 3 decimal places Making the coefﬁcient of y 2 unity gives: y2 +

i.e. and

1 or −3 are the roots of the equation 2 2 2x + 5x = 3 Hence x =

Problem 7. Solve 2x 2 + 9x + 8 = 0, correct to 3 signiﬁcant ﬁgures, by ‘completing the square’

3.5 1.75 y− =0 4.6 4.6

and rearranging gives: y 2 +

3.5 1.75 y= 4.6 4.6

Adding to both sides (half the coefﬁcient of y)2 gives: 3.5 2 1.75 3.5 2 3.5 y+ + y + = 4.6 9.2 4.6 9.2

Solve the simple equation. Thus 5 7 x =− ± 4 4 5 7 2 1 x =− + = = 4 4 4 2 5 7 12 x = − − = − = −3 4 4 4

87

2

The LHS is now a perfect square, thus: 3.5 2 = 0.5251654 y+ 9.2 Taking the square root of both sides gives: 3.5 √ = 0.5251654 = ±0.7246830 9.2 3.5 ± 0.7246830 Hence, y = − 9.2 i.e y = 0.344 or −1.105 y+

Section 1

Solving quadratic equations

Section 1

88 Engineering Mathematics Now try the following exercise Exercise 43

Further problems on solving quadratic equations by ‘completing the square’

Solve the following equations by completing the square, each correct to 3 decimal places. 1.

x 2 + 4x + 1 =0

[−3.732, −0.268]

2. 2x 2 + 5x − 4 =0

[−3.137, 0.637]

3. 3x 2 − x − 5 =0

[1.468, −1.135]

4. 5x 2 − 8x + 2 =0

[1.290, 0.310]

5. 4x 2 − 11x + 3 = 0

[2.443, 0.307]

11.4 Solution of quadratic equations by formula Let the general form of a quadratic equation be given by: ax 2 + bx + c = 0 where a, b and c are constants. Dividing ax2 + bx + c = 0 by a gives: b c x2 + x + = 0 a a Rearranging gives: b c x2 + x = − a a Adding to each side of the equation the square of half the coefﬁcient of the terms in x to make the LHS a perfect square gives: 2 2 b b c b = − x2 + x + a 2a 2a a Rearranging gives: b2 c b2 − 4ac b 2 = 2− = x+ a 4a a 4a 2 Taking the square root of both sides gives: √ b 2 − 4ac ± b 2 − 4ac b = = x+ 2a 4a 2 2a √ b 2 − 4ac b Hence x = − ± 2a 2a

√ −b ± b 2 − 4ac i.e. the quadratic formula is: x= 2a (This method of solution is ‘completing the square’ – as shown in Section 10.3.) Summarising: if ax 2 + bx + c = 0 then

−b ± b2 − 4ac x= 2a

This is known as the quadratic formula. Problem 9. Solve (a) x 2 + 2x − 8 =0 and (b) 3x 2 − 11x − 4 = 0 by using the quadratic formula (a)

Comparing x 2 + 2x − 8 =0 with ax2 + bx + c = 0 gives a = 1, b = 2 and c = −8. Substituting these values into the quadratic formula √ −b ± b2 − 4ac gives x= 2a −2 ± 22 − 4(1)(−8) x= 2(1) √ √ −2 ± 4 + 32 −2 ± 36 = = 2 2 −2 ± 6 −2 + 6 −2 − 6 = = or 2 2 2 −8 4 = −4 (as in Hence x = = 2 or 2 2 Problem 1(a)).

(b) Comparing 3x 2 − 11x− 4 =0 with ax2 + bx + c = 0 gives a = 3, b = −11 and c = −4. Hence, −(−11) ± (−11)2 − 4(3)(−4) x= 2(3) √ √ −11 ± 121 + 48 11 ± 169 = = 6 6 11 ± 13 11 + 13 11 − 13 = = or 6 6 6 −2 1 24 =− Hence x = = 4 or 6 6 3 Problem 1(b)).

(as

Problem 10. Solve 4x 2 + 7x + 2 = 0 giving the roots correct to 2 decimal places

in

89

Comparing 4x 2 + 7x + 2 = 0 with ax2 + bx + c = 0 gives a = 4, b = 7 and c = 2. Hence, −7 ± 72 − 4(4)(2) x= 2(4) √ −7 ± 17 −7 ± 4.123 = = 8 8 −7 ± 4.123 −7 − 4.123 = or 8 8

Since the total surface area = 2.0 m2 and the height h = 82 cm or 0.82 m, then

Hence, x = −0.36 or −1.39, correct to 2 decimal places.

Using the quadratic formula:

2.0 = 2πr(0.82) + 2πr 2 i.e. 2πr 2 + 2πr(0.82) −2.0 = 0 Dividing throughout by 2π gives: r 2 + 0.82r −

r=

Now try the following exercise

1 =0 π

1 −0.82 ± (0.82)2 − 4(1) − π 2(1) √

=

Exercise 44 Further problems on solving quadratic equations by formula Solve the following equations by using the quadratic formula, correct to 3 decimal places. 1.

2x 2 + 5x − 4 =0

[0.637, −3.137]

2.

5.76x 2 + 2.86x − 1.35 =0

[0.296, −0.792]

3.

2x 2 − 7x + 4 =0

4.

3 4x + 5 = x

5.

(2x + 1) =

= 0.2874

11.5 Practical problems involving quadratic equations There are many practical problems where a quadratic equation has ﬁrst to be obtained, from given information, before it is solved.

= 0.5748 m

Total surface area of a cylinder = curved surface area + 2 circular ends (from Chapter 20) = 2πrh = 2πr 2 (where r = radius and h = height)

or

57.5 cm

correct to 3 signiﬁcant ﬁgures Problem 12. The height s metres of a mass projected vertically upward at time t seconds is 1 s = ut − gt 2. Determine how long the mass will 2 take after being projected to reach a height of 16 m (a) on the ascent and (b) on the descent, when u =30 m/s and g = 9.81 m/s2 1 When height s = 16 m, 16 =30t − (9.81)t 2 2 i.e.

Problem 11. Calculate the diameter of a solid cylinder which has a height of 82.0 cm and a total surface area of 2.0 m2

−1.1074

= 2 × 0.2874

[2.781, 0.719]

[3.608, −1.108]

or

Thus the radius r of the cylinder is 0.2874 m (the negative solution being neglected). Hence the diameter of the cylinder

[0.443, −1.693] 5 x −3

−0.82 ± 1.9456 −0.82 ± 1.3948 = 2 2

4.905t 2 − 30t + 16 = 0

Using the quadratic formula: −(−30) ± (−30)2 − 4(4.905)(16) t= 2(4.905) √ 30 ± 586.1 30 ± 24.21 = = 9.81 9.81 = 5.53 or 0.59 Hence the mass will reach a height of 16 m after 0.59 s on the ascent and after 5.53 s on the descent.

Section 1

Solving quadratic equations

Section 1

90 Engineering Mathematics Problem 13. A shed is 4.0 m long and 2.0 m wide. A concrete path of constant width is laid all the way around the shed. If the area of the path is 9.50 m2 calculate its width to the nearest centimetre Figure 11.1 shows a plan view of the shed with its surrounding path of width t metres. Area of path = 2(2.0 × t ) + 2t (4.0 + 2t ) i.e. 9.50 = 4.0t + 8.0t + 4t 2 or 4t 2 + 12.0t − 9.50 = 0 t t

2.0 m

SHED

Figure 11.1

Hence t =

−(12.0) ±

(12.0)2 − 4(4)(−9.50) 2(4)

√ −12.0 ± 296.0 = 8 −12.0 ± 17.20465 = 8 Hence t = 0.6506 m or −3.65058 m

Neglecting the negative result which is meaningless, the width of the path, t =0.651 m or 65 cm, correct to the nearest centimetre. Problem 14. If the total surface area of a solid cone is 486.2 cm2 and its slant height is 15.3 cm, determine its base diameter From Chapter 20, page 163, the total surface area A of a solid cone is given by: A = πrl + πr 2 where l is the slant height and r the base radius. If A = 482.2 and l = 15.3, then 482.2 = πr(15.3) + πr 2 i.e. or

πr 2 + 15.3πr − 482.2 = 0 r 2 + 15.3r −

482.2 =0 π

−15.3 ± r= −15.3 ±

(15.3)2 − 4 √

−482.2 π

2

848.0461 2 −15.3 ± 29.12123 = 2 Hence radius r = 6.9106 cm (or −22.21 cm, which is meaningless, and is thus ignored). Thus the diameter of the base = 2r = 2(6.9106) = 13.82 cm =

Now try the following exercise (4.0 ⫹ 2t )

4.0 m

Using the quadratic formula,

Exercise 45

Further practical problems involving quadratic equations

1. The angle a rotating shaft turns through in t seconds is given by: 1 θ = ωt + αt 2 . Determine the time taken to 2 complete 4 radians if ω is 3.0 rad/s and α is [1.191 s] 0.60 rad/s2 2. The power P developed in an electrical circuit is given by P = 10I − 8I 2 , where I is the current in amperes. Determine the current necessary to produce a power of 2.5 watts in the circuit [0.345 A or 0.905 A] 3. The sag l metres in a cable stretched between two supports, distance x m apart is given by: 12 l = + x. Determine the distance between x supports when the sag is 20 m [0.619 m or 19.38 m] 4. The acid dissociation constant K a of ethanoic acid is 1.8 × 10−5 mol dm−3 for a particular solution. Using the Ostwald dilution law x2 determine x, the degree of Ka = v(1 − x) ionization, given that v = 10 dm3 [0.0133] 5. A rectangular building is 15 m long by 11 m wide. A concrete path of constant width is laid all the way around the building. If the area of the path is 60.0 m2 , calculate its width correct to the neareast millimetre [1.066 m]

2x 2 + x − 3 = 0

or 6.

7.

8.

9.

The total surface area of a closed cylindrical container is 20.0 m2 . Calculate the radius of the cylinder if its height is 2.80 m [86.78 cm] The bending moment M at a point in a beam 3x(20 − x) is given by M = where x metres is 2 the distance from the point of support. Determine the value of x when the bending moment is 50 Nm [1.835 m or 18.165 m]

Factorising gives:

(2x + 3)(x − 1) = 0

i.e.

x =−

3 or x = 1 2

In the equation y = 6x − 7 when

3 −3 x =− , y =6 − 7 = −16 2 2

and when

x = 1, y = 6 − 7 = −1

A tennis court measures 24 m by 11 m. In the layout of a number of courts an area of ground must be allowed for at the ends and at the sides of each court. If a border of constant width is allowed around each court and the total area of the court and its border is 950 m2 , ﬁnd the width of the borders [7 m]

[Checking the result in y = 5x − 4 − 2x 2 :

Two resistors, when connected in series, have a total resistance of 40 ohms. When connected in parallel their total resistance is 8.4 ohms. If one of the resistors has a resistance Rx ohms: (a) show that Rx2 − 40Rx + 336 =0 and (b) calculated the resistance of each [(b) 12 ohms, 28 ohms]

as above; and when x = 1, y = 5 −4 − 2 = −1 as above.] Hence the simultaneous solutions occur when

3 3 2 3 −4−2 − when x = − , y = 5 − 2 2 2 =−

15 9 − 4 − = −16 2 2

3 x = − , y = −16 2 and when x = 1, y = −1 Now try the following exercise

11.6 The solution of linear and quadratic equations simultaneously Sometimes a linear equation and a quadratic equation need to be solved simultaneously. An algebraic method of solution is shown in Problem 15; a graphical solution is shown in Chapter 31, page 281. Problem 15. Determine the values of x and y which simultaneously satisfy the equations: y = 5x − 4 − 2x 2 and y = 6x − 7 For a simultaneous solution the values of y must be equal, hence the RHS of each equation is equated. Thus 5x − 4 −2x 2 = 6x − 7 Rearranging gives: 5x 2 − 4 − 2x 2 − 6x + 7 = 0 i.e.

91

−x + 3 − 2x 2 = 0

Exercise 46

Further problems on solving linear and quadratic equations simultaneously

In Problems 1 to 3 determine the solutions of the simulations equations. 1.

y = x2 + x + 1 y=4−x [x = 1, y = 3 and x = −3, y = 7]

2.

y = 15x 2 + 21x − 11 y = 2x − 1 1 2 1 2 x = , y = − and x = −1 , y = −4 5 5 3 3

3.

2x 2 + y = 4 + 5x x + y=4 [x = 0, y = 4 and x = 3, y = 1]

Section 1

Solving quadratic equations

Chapter 12

Inequalities 12.1 Introduction in inequalities An inequality is any expression involving one of the symbols < , > ≤ or ≥ p < q means p is less than q p > q means p is greater than q p ≤q means p is less than or equal to q p ≥q means p is greater than or equal to q

Some simple rules (i) When a quantity is added or subtracted to both sides of an inequality, the inequality still remains.

12.2

The solution of some simple inequalities, using only the rules given in Section 12.1, is demonstrated in the following worked problems. Problem 1. Solve the following inequalities: (a) 3 + x > 7 (b) 3t < 6 p (c) z −2 ≥ 5 (d) ≤ 2 3 (a)

For example, if p < 3 then

p + 2 < 3 + 2 (adding 2 to both sides)

and

p − 2 < 3 − 2 (subtracting 2 from both sides)

(ii) When multiplying or dividing both sides of an inequality by a positive quantity, say 5, the inequality remains the same. For example,

Hence, all values of x greater than 4 satisfy the inequality. (b) Dividing both sides of the inequality: 3t < 6 by 3 gives: 6 3t < , i.e. t < 2 3 3 Hence, all values of t less than 2 satisfy the inequality. (c)

1 p < if p > 1 then − 3 p < −3 and −3 −3 (Note >has changed to 7 gives: 3 + x − 3 > 7 − 3, i.e. x > 4

p 4 if p > 4 then 5 p > 20 and > 5 5 (iii) When multiplying or dividing both sides of an inequality by a negative quantity, say −3, the inequality is reversed. For example,

Simple inequalities

Adding 2 to both sides of the inequality z − 2 ≥5 gives: z − 2 + 2 ≥ 5 + 2, i.e. z ≥ 7

Hence, all values of z greater than or equal to 7 satisfy the inequality. p (d) Multiplying both sides of the inequality ≤ 2 by 3 3 gives: p (3) ≤ (3)2, i.e. p ≤ 6 3 Hence, all values of p less than or equal to 6 satisfy the inequality.

Problem 2. Solve the inequality: 4x + 1 > x + 5 Subtracting 1 from both sides of the inequality: 4x + 1 > x + 5 gives: 4x > x + 4 Subtracting x from both sides of the inequality: 4x > x + 4 gives: 3x > 4 Dividing both sides of the inequality: 3x > 4 by 3 gives: x>

4 3

4 satisfy the Hence all values of x greater than 3 inequality: 4x + 1 > x + 5 Problem 3. Solve the inequality: 3 −4t ≤ 8 +t Subtracting 3 from both sides of the inequality: 3 − 4t ≤8 + t gives: −4t ≤ 5 + t Subtracting t from both sides of the inequality: −4t ≤ 5 +t gives: −5t ≤ 5 Dividing both sides of the inequality −5t ≤ 5 by −5 gives: t ≥ −1 (remembering to reverse the inequality) Hence, all values of t greater than or equal to −1 satisfy the inequality. Now try the following exercise Exercise 47 Further problems on simple inequalities Solve the following inequalities: 1.

(a) 3t > 6

(b) 2x < 10 [(a) t > 2

x > 1.5 (b) x + 2 ≥5 2 [(a) x > 3 (b) x ≥ 3]

2.

(a)

3.

(a) 4t − 1 ≤ 3

4.

(b) x < 5]

(b) 5 − x ≥ − 1 [(a) t ≤ 1 (b) x ≤ 6]

7 −2k (a) ≤ 1 (b) 3z + 2 > z + 3 4 3 1 (a) k ≥ (b) z > 2 2

5.

93

(a) 5 − 2y ≤ 9 + y (b) 1 − 6x ≤ 5 + 2x 4 1 (a) y ≥ − (b) x ≥ − 3 2

12.3 Inequalities involving a modulus The modulus of a number is the size of the number, regardless of sign. Vertical lines enclosing the number denote a modulus. For example, |4| =4 and |−4| =4 (the modulus of a number is never negative), The inequality: |t | 3 means all numbers whose actual size, regardless of sign, is greater than 3, i.e. any value greater than 3 and any value less than −3. Thus |x| > 3 means x > 3 and x < −3. Inequalities involving a modulus are demonstrated in the following worked problems. Problem 4. Solve the following inequality: |3x + 1| 0 or (ii) t + 1 0 then 3t > 6 and t > 2 Both of the inequalities t > −1 and t > 2 are only true when t > 2, t +1 is positive when t > 2 i.e. the fraction 3t − 6 (ii) If t + 1 2 or t < −1 Summarising, 3t − 6

[k ≥ 4 and k ≤ −2] Problem 8.

12.4

Inequalities involving quotients

p p If > 0 then must be a positive value. q q p to be positive, either p is positive and q is For q positive or p is negative and q is negative. + − i.e. = + and = + + − p p If < 0 then must be a negative value. q q p to be negative, either p is positive and q is For q negative or p is negative and q is positive. + − i.e. = − and = − − + This reasoning is used when solving inequalities involving quotients as demonstrated in the following worked problems. Problem 7.

Solve the inequality:

t +1 >0 3t − 6

t +1 t +1 > 0 then must be positive. Since 3t − 6 3t − 6

Since

Solve the inequality:

2x + 3 ≤1 x +2

2x + 3 2x + 3 ≤ 1 then − 1 ≤0 x +2 x +2

i.e.

2x + 3 x + 2 − ≤ 0, x +2 x +2

i.e.

2x + 3 − (x + 2) x +1 ≤ 0 or ≤0 x +2 x +2

For

x +1 to be negative or zero, x +2 either (i) x + 1 ≤0 and x + 2 > 0 or (ii) x + 1 ≥0 and x + 2 < 0

(i) If x + 1 ≤ 0 then x ≤ −1 and if x + 2 >0 then x > −2. (Note that >is used for the denominator, not ≥; a zero denominator gives a value for the fraction which is impossible to evaluate.) x +1 ≤ 0 is true when x is x +2 greater than −2 and less than or equal to −1, which may be written as −2 < x ≤ −1 Hence, the inequality

(ii) If x + 1 ≥ 0 then x ≥ −1 and if x + 2 9 when t > 3 or t < −3 This demonstrates the general rule: if x2 > k

then

x>

√ k

or

√ x 4 Exercise 49 Further problems on inequalities involving quotients Solve the following inequalitites: 1. 2. 3. 4.

x +4 ≥0 6 − 2x 2t + 4 >1 t −5 3z − 4 ≤2 z +5 2−x ≥4 x +3

[−4 ≤ x < 3]

From the general√rule stated √above in equation (1): if x 2 > 4 then x > 4 or x < − 4 i.e. the inequality: x 2 > 4 is satisﬁed when x > 2 or x < −2 Problem 11. Solve the inequality: (2z +1)2 > 9

[t > 5 or t < −9] [−5 < z ≤14] [−3 < x ≤ −2]

From equation (1), if (2z +1)2 > 9 then √ √ 2z +1 > 9 or 2z + 1 3 or 2z + 1 2 or 2z < −4, z > 1 or z < −2

Problem 12. Solve the inequality: t 2 < 9

12.5 Inequalities involving square functions The following two general rules apply when inequalities involve square functions: √ √ (1) (i) if x2 > k then x > k or x < − k √ √ (2) (ii) if x2 > k then − k < x < k These rules are demonstrated in the following worked problems. Problem 9. Solve the inequality: t > 9 2

Since t 2 > 9 then t 2 − 9 > 0, i.e. (t + 3)(t − 3) > 0 by factorising. For (t + 3)(t − 3) to be positive, either (i) (t + 3) > 0 and (t − 3) >0 or (ii) (t + 3) < 0 and (t − 3) < 0 (i) If (t + 3) > 0 then t > −3 and if (t − 3) > 0 then t >3 Both of these are true only when t > 3 (ii) If (t + 3) < 0 then t < −3 and if (t − 3) < 0 then t 0 then t > −3 and if (t − 3) < 0 then t −3 and t < 3 which may be written as: −3 < t < 3 (ii) If (t + 3) < 0 then t < −3 and if (t − 3) > 0 then t >3 It is not possible to satisfy both t < −3 and t >3, thus no values of t satisﬁes (ii). Summarising, t 2 < 9 when −3 < t < 3 which means that all values of t between −3 and +3 will satisfy the inequality. This demonstrates the general rule: √ √ if x2 < k then − k < x < k

(2)

Problem 13. Solve the inequality: x 2 < 4 From the general above in equation (2): if √ rule stated √ x 2 < 4 then − 4 < x < 4

Section 1

Inequalities

Section 1

96 Engineering Mathematics i.e. the inequality: x 2 < 4 is satisﬁed when: −2 < x < 2 Problem 14.

Solve the inequality: (y − 3)2 ≤ 16

√ √ From equation (2), − 16 ≤(y − 3) ≤ 16 −4 ≤(y − 3) ≤4

i.e.

x 2 + 4x − 7 ≡ (x + 2)2 − 7 − 22 ≡ (x + 2)2 − 11 Similarly, x 2 + 6x − 5 ≡ (x + 3)2 − 5 − 32 ≡ (x − 3)2 − 14

3 −4 ≤ y ≤ 4 +3,

from which,

−1 ≤ y ≤ 7

i.e.

Further problems on inequalities involving square functions

Solve the following inequalities: 1.

z 2 > 16

[z >4 or z < −4]

2.

z 2 < 16

[−4 < z < 4] √ [x ≥ 3 or x ≤− 3] √

3. 2x 2 ≥ 6 4. 3k 2 − 2 ≤10

[−2 ≤ k ≤ 2]

5.

(t − 1)2 ≤ 36

[−5 ≤t ≤ 7]

6.

(t − 1)2 ≥ 36

[t ≥ 7 or t ≤ −5]

7. 7 − 3y 2 ≤ −5

8. (4k + 5)2 > 9

Solving quadratic inequalities is demonstrated in the following worked problems. Problem 15.

Now try the following exercise Exercise 50

For example, x 2 + 4x − 7 does not factorise; completing the square gives:

[y ≥ 2 or y ≤ −2] 1 k > − or k < −2 2

Since x 2 + 2x − 3 >0 then (x − 1)(x + 3) > 0 by factorising. For the product (x − 1)(x + 3) to be positive, either (i) (x − 1) >0 and (x + 3) > 0 or (ii) (x − 1) < 0 and (x + 3) < 0 (i) Since (x − 1) > 0 then x > 1 and since (x + 3) > 0 then x > −3 Both of these inequalities are satisﬁed only when x>1 (ii) Since (x − 1) < 0 then x < 1 and since (x + 3) < 0 then x < −3 Both of these inequalities are satisﬁed only when x < −3 Summarising, x 2 + 2x − 3 >0 is satisﬁed when either x > 1 or x < −3 Problem 16.

12.6

Quadratic inequalities

Inequalities involving quadratic expressions are solved using either factorisation or ‘completing the square’. For example,

Solve the inequality: x 2 + 2x − 3 >0

Solve the inequality: t 2 − 2t −8 < 0

Since t 2 − 2t − 8 0 and (t + 2) < 0 or (ii) (t − 4) < 0 and (t + 2) > 0

and 6x 2 + 7x − 5 is factorised as (2x − 1)(3x + 5)

(i) Since (t − 4) > 0 then t > 4 and since (t + 2) < 0 then t < −2 It is not possible to satisfy both t > 4 and t < −2, thus no values of t satisﬁes the inequality (i)

If a quadratic expression does not factorise, then the technique of ‘completing the square’ is used. In general, the procedure for x 2 +bx + c is:

(ii) Since (t − 4) < 0 then t < 4 and since (t + 2) > 0 then t > −2 Hence, (ii) is satisﬁed when −2 3 or x < −2]

2.

t 2 + 2t − 8 ≤ 0

3.

2x 2 + 3x − 2 < 0

[−4 ≤t ≤ 2] 1 −2 < x < 2

4.

y 2 − y − 20 ≥ 0

[y ≥ 5 or y ≤ −4]

5.

z 2 + 4z + 4 ≤4

[−4 ≤ z ≤0]

6.

x 2 + 6x − 6 ≤0 √ √ (− 3 − 3) ≤ x ≤ ( 3 − 3)

7.

t 2 − 4t − 7 ≥ 0 √ √ [t ≥ ( 11 + 2) or t ≤ (2 − 11)]

8.

k 2 + k − 3 ≥0 13 1 13 1 k≥ or k ≤ − − − 4 2 4 2

Section 1

Inequalities

Chapter 13

Logarithms 13.1 Introduction to logarithms With the use of calculators ﬁrmly established, logarithmic tables are now rarely used for calculation. However, the theory of logarithms is important, for there are several scientiﬁc and engineering laws that involve the rules of logarithms. From Chapter 5, we know that: 16 = 24 . The number 4 is called the power or the exponent or the index. In the expression 24 , the number 2 is called the base. In another example, we know that: 64 = 82 . In this example, 2 is the power, or exponent, or index. The number 8 is the base.

What is a logarithm? Consider the expression 16 = 24 . An alternative, yet equivalent, way of writing this expression is: log2 16 = 4. This is stated as ‘log to the base 2 of 16 equals 4’. We see that the logarithm is the same as the power or index in the original expression. It is the base in the original expression which becomes the base of the logarithm. The two statements: 16 = 24 and log2 16 = 4 are equivalent. If we write either of them, we are automatically implying the other. In general, if a number y can be written in the form a x , then the index x is called the ‘logarithm of y to the base of a’, i.e.

if y = ax then x = loga y

In another example, if we write down that 64 = 82 then the equivalent statement using logarithms is: log8 64 = 2 DOI: 10.1016/B978-0-08-096562-8.00013-4

In another example, if we write down that: log3 27 = 3 then the equivalent statement using powers is: 33 = 27 So the two sets of statements, one involving powers and one involving logarithms, are equivalent.

Common logarithms From above, if we write down that: 1000 = 103 , then 3 = log10 1000. This may be checked using the ‘log’ button on your calculator. Logarithms having a base of 10 are called common logarithms and log10 is usually abbreviated to lg. The following values may be checked by using a calculator: lg 27.5 = 1.4393 . . ., lg 378.1 = 2.5776 . . . and lg 0.0204 = −1.6903 . . .

Napierian logarithms Logarithms having a base of e (where ‘e’ is a mathematical constant approximately equal to 2.7183) are called hyperbolic, Napierian or natural logarithms, and loge is usually abbreviated to ln. The following values may be checked by using a calculator: ln 3.65 = 1.2947 . . ., ln 417.3 = 6.0338 . . . and ln 0.182 = −1.7037 . . . More on Napierian logarithms is explained in Chapter 14. Here are some worked problems to help you understand logarithms.

Logarithms

Let x = log3 9 then 3 x = 9 from the deﬁnition of a logarithm, i.e. 3x = 32 from which, x = 2 log 3 9 = 2

Hence,

Hence,

log3

1 = −4 81

Section 1

Problem 1. Evaluate: log3 9

Problem 7. Solve the equation: lg x = 3 If lg x = 3 then log10 x = 3 x = 103 i.e. x = 1000

and Problem 2. Evaluate: log10 10 Let x = log10 10 then 10 x = 10 from the deﬁnition of a logarithm, x 1 10 = 10 from which, x = 1 i.e. Hence,

Problem 8. Solve the equation: log2 x = 5 If log2 x = 5 then x = 25 = 32 Problem 9. Solve the equation: log5 x = −2

log10 10 = 1 (which may be checked by a calculator)

If log5 x = −2 then x = 5−2 =

Problem 3. Evaluate: log16 8 Let x = log16 8 then 16x = 8 from the deﬁnition of a logarithm, (24 )x = 23 , i.e. 24x = 23 from the laws of indices, 3 from which, 4x = 3 and x = 4 3 Hence, log16 8 = 4 Problem 4. Evaluate: lg 0.001 Let x = lg 0.001 = log10 0.001 then 10x = 0.001 i.e.

10 x = 10−3 from which x = −3

Hence, lg 0.001 = −3 (which may be checked by a calculator) Problem 5. Evaluate: ln e Let x = ln e = loge e then i.e. Hence,

x

ex

= e,

1

e = e from which x = 1 ln e = 1 (which may be checked by a calculator)

1 Problem 6. Evaluate: log3 81 1 1 1 then 3 x = = 4 = 3−4 from which, Let x = log3 81 81 3 x = −4

1 1 = 52 25

Now try the following exercise Exercise 52

i.e.

Further problems on laws of logarithms

In Problems 1 to 11, evaluate the given expression: 1. log10 10 000

[4]

3. log5 125

[3] 1 3

5. log8 2 7. lg 100

9. log4 8 11. ln e2

2. log2 16 4. log2

[4]

1 8

[−3]

6. log7 343

[3]

[2] 8. lg 0.01 1 1 10. log27 3 2

[−2] 1 3

[2]

In Problems 12 to 18 solve the equations: 12. log10 x = 4

[10 000]

13. lg x = 5

[100 000]

14. log3 x = 2 15. log4 x = −2 16. lg x = −2 17. log8 x = − 18. ln x = 3

4 3

1 2

99

[9] 1 32

[0.01] 1 16 3 e

Section 1

100 Engineering Mathematics

13.2

Laws of logarithms

There are three laws of logarithms, which apply to any base:

16 log 16 − log 2 = 2 by the second law of logarithms = log 8

(i) To multiply two numbers: log (A × B) = log A + log B

Problem 12. Write 2 log 3 as the logarithm of a single number

The following may be checked by using a calculator:

2 log 3 = log 32 by the third law of logarithms

lg 10 = 1 Also, lg 5 + lg 2 = 0.69897 . . . + 0.301029 . . . = 1. Hence, lg (5 × 2) = lg 10 = lg 5 + lg 2. (ii) To divide two numbers: A log = log A − log B B

= log 9 1 Problem 13. Write log 25 as the logarithm of a 2 single number 1 1 log 25 = log 25 2 2 by the third law of logarithms √ = log 25 = log 5

The following may be checked using a calculator: 5 = ln 2.5 = 0.91629 . . . ln 2 Also, ln 5 − ln 2 = 1.60943 . . . − 0.69314 . . . = 0.91629 . . . 5 Hence, ln = ln 5 − ln 2. 2 (iii) To raise a number to a power: log An = n log A The following may be checked using a calculator: lg 5 = lg 25 = 1.39794 . . .

Problem 14.

Simplify: log 64 − log 128 + log 32

64 = 26 , 128 = 27 and 32 = 25 Hence,

log64 − log 128 + log32 = log 26 − log 27 + log 25 = 6 log2 − 7 log2 + 5 log 2 by the third law of logarithms = 4 log 2

2

Also, Hence,

2 lg 5 = 2 × 0.69897 . . . = 1.39794 . . . lg 52 = 2 lg 5.

Here are some worked problems to help you understand the laws of logarithms.

1 1 log 16 + log 27 − 2 log5 2 3 as the logarithm of a single number

Problem 15.

Write

1 1 log 16 + log 27 − 2 log5 2 3 1

Problem 10. Write log4 + log 7 as the logarithm of a single number log 4 + log 7 = log (7 × 4) by the ﬁrst law of logarithms = log 28 Problem 11. Write log 16 − log 2 as the logarithm of a single number

1

= log 16 2 + log 27 3 − log 52 by the third law of logarithms √ √ 3 = log 16 + log 27 − log 25 by the laws of indices = log 4 + log 3 − log25 4×3 = log 25 by the ﬁrst and second laws of logarithms 12 = log = log 0.48 25

Problem 16. Write (a) log 30 (b) log 450 in terms of log2, log 3 and log 5 to any base (a)

log 30 = log(2 × 15) = log(2 × 3 × 5) = log 2 + log 3 + log 5 by the ﬁrst law of logarithms

Problem 19. Solve the equation: log(x − 1) + log(x + 8) = 2 log(x + 2) LHS = log(x − 1) + log(x + 8) = log(x − 1)(x + 8) from the ﬁrst law of logarithms = log(x 2 + 7x − 8) = 2 log(x + 2) = log(x + 2)2 from the third law of logarithms

RHS (b) log 450 = log(2 × 225) = log(2 × 3 × 75)

= log(x 2 + 4x + 4)

= log(2 × 3 × 3 × 25) = log(2 × 32 × 52 )

i.e.

= log 2 + log 32 + log 52 by the ﬁrst law of logarithms,

Hence,

log 450 = log 2 + 2 log 3 + 2 log 5 by the third law of logarithms

i.e.

7x − 8 = 4x + 4,

i.e.

3x = 12

and

x=4

√ 4 8× 5 in terms of Problem 17. Write log 81 log2, log 3 and log 5 to any base

log(x 2 + 7x − 8) = log(x 2 + 4x + 4) x 2 + 7x − 8 = x 2 + 4x + 4,

from which,

Problem 20. Solve the equation:

√ 4 √ 8× 5 4 log = log 8 + log 5 − log 81 81 by the ﬁrst and second laws of logarithms 1

i.e.

√ 8× 4 5 1 log = 3 log 2 + log 5 − 4 log 3 81 4 by the third law of logarithms

log 25 − log125 +

1 log625 2

3 log5 log25 − log 125 +

1 log 625 2

3 log 5 =

=

Hence, becomes

1 log 4 = log x 2 √ log 4 = log x,

i.e. from which,

Problem 18. Evaluate:

log52 − log 53 +

1 log54 2

3 log5 2 log5 − 3 log5 + 3 log5

4 log5 1 log5 1 2 = = 3 log5 3

1 log 4 = log x 2

1 1 log 4 = log 4 2 2 from the third law of logarithms √ = log 4 from the laws of indices

= log 23 + log 5 4 − log 34 by the laws of indices,

101

log 2 = log x 2 = x,

i.e. the solution of the equation is: x = 2 Problem 21. Solve the equation: log(x 2 − 3) − log x = log 2

x2 − 3 x from the second law of logarithms 2 x −3 = log 2 log x

log(x 2 − 3) − log x = log

Hence, from which, Rearranging gives:

x −3 =2 x x 2 − 3 = 2x

Section 1

Logarithms

Section 1

102 Engineering Mathematics x 2 − 2x − 3 = 0

and Factorising gives:

(x − 3)(x + 1) = 0

16.

x = 3 or x = −1

from which,

Further problems on laws of logarithms

1.

log 2 + log 3

[log 6]

2.

log 3 + log 5

[log15]

3.

log 3 + log 4 − log 6

[log 2]

4.

log 7 + log 21 − log49

[log 3]

5.

2 log2 + log 3

6.

2 log2 + 3 log 5

1 log 81 + log36 2 1 1 log 8 − log81 + log 27 8. 3 2 1 log4 − 2 log 3 + log45 9. 2 1 log16 + 2 log3 − log 18 10. 4 11. 2 log2 + log 5 − log 10

18. log 2t 3 − log t = log 16 + logt 2 logb 2 − 3 logb

= log8b − log 4b

20. log(x + 1) + log(x − 1) = log 3

In Problems 1 to 11, write as the logarithm of a single number:

2 log5 −

17. log x 4 − log x 3 = log 5x − log 2x [x = 2.5]

19.

Now try the following exercise

7.

[1.5]

Solve the equations given in Problems 17 to 22:

x = −1 is not a valid solution since the logarithm of a negative number has no real root. Hence, the solution of the equation is: x = 3

Exercise 53

1 log 9 − log 3 + log 81 3 2 log3

[log12]

[t = 8] [b = 2] [x = 2]

1 log 27 = log(0.5a) 3

[a = 6]

22. log(x 2 − 5) − log x = log4

[x = 5]

21.

13.3

Indicial equations

The laws of logarithms may be used to solve certain equations involving powers — called indicial equations. For example, to solve, say, 3 x = 27, logarithms to base of 10 are taken of both sides,

[log 500]

i.e.

log10 3 x = log10 27

[log100]

and

x log10 3 = log10 27 by the third law of logarithms.

[log 6] x=

Rearranging gives [log 10]

=

[log1 = 0] [log 2]

Simplify the expressions given in Problems 12 to 14: 12. log 27 − log9 + log 81 [log 243 or log 35 or 5 log3]

log10 27 log10 3 1.43136 . . . =3 0.4771 . . .

which may be readily checked. (Note, (log8/ log2) is not equal to lg (8/2).) Problem 22. Solve the equation 2 x = 3, correct to 4 signiﬁcant ﬁgures

13. log 64 + log 32 − log 128 [log 16 or log24 or 4 log2]

Taking logarithms to base 10 of both sides of 2 x = 3 gives:

14. log 8 − log 4 + log 32 [log 64 or log 26 or 6 log2]

log10 2 x = log10 3

Evaluate the expressions given in Problems 15 and 16: 1 1 log16 − log 8 3 15. 2 [0.5] log 4

x log10 2 = log10 3

i.e.

Rearranging gives: x=

log10 3 0.47712125 . . . = = 1.585 log10 2 0.30102999 . . . correct to 4 signiﬁcant ﬁgures.

Problem 23. Solve the equation 2x+1 = 32x−5 correct to 2 decimal places Taking logarithms to base 10 of both sides gives:

7.

x −0.25 = 0.792

8.

0.027 x = 3.26

9.

The decibel gain n of an ampliﬁer is given P2 by: n =10 log10 where P1 is the power P1 input and P2 is the power output. Find the P2 when n = 25 decibels. power gain P1 [316.2]

log10 2 x+1 = log10 32x−5 (x + 1) log10 2 = (2x − 5) log10 3

i.e.

x log10 2 + log10 2 = 2x log10 3 − 5 log10 3 x(0.3010) + (0.3010) = 2x(0.4771) − 5(0.4771) i.e.

[2.542] [−0.3272]

0.3010x + 0.3010 = 0.9542x − 2.3855 2.3855 + 0.3010 = 0.9542x − 0.3010x

Hence

2.6865 = 0.6532x x=

from which

2.6865 = 4.11 0.6532 correct to 2 decimal places.

13.4 Graphs of logarithmic functions A graph of y = log10 x is shown in Fig. 13.1 and a graph of y = loge x is shown in Fig. 13.2. Both are seen to be of similar shape; in fact, the same general shape occurs for a logarithm to any base.

Problem 24. Solve the equation x 3.2 = 41.15, correct to 4 signiﬁcant ﬁgures

y

0.5

Taking logarithms to base 10 of both sides gives: log10 x 3.2 = log10 41.15 3.2 log10 x = log10 41.15 log10 x =

Hence

1

x y 5 log10x

20.5

log10 41.15 = 0.50449 3.2

2 3

3 2

1

x 0.5

0.2

0.1

0.48 0.30 0 2 0.30 2 0.70 2 1.0

21.0

Thus x =antilog 0.50449 =100.50449 = 3.195 correct to 4 signiﬁcant ﬁgures.

Figure 13.1

Now try the following exercise Exercise 54 Indicial equations

y

Solve the following indicial equations for x, each correct to 4 signiﬁcant ﬁgures:

2

1

1.

3x = 6.4

[1.690]

2.

2x = 9

[3.170]

3.

2 x−1 = 32x−1

[0.2696]

4.

x 1.5 = 14.91

[6.058]

5.

25.28 =4.2

[2.251]

6.

42x−1 = 5x+2

x

0 21 22

[3.959] Figure 13.2

1

2

3

4

5

6

x

x 6 5 4 3 2 1 0.5 0.2 0.1 y 5 loge x 1.79 1.61 1.39 1.10 0.69 0 20.69 21.61 22.30

Section 1

103

Logarithms

Section 1

104 Engineering Mathematics In general, with a logarithm to any base a, it is noted that: (i) loga 1 =0 Let loga = x, then a x = 1 from the deﬁnition of the logarithm. If a x = 1 then x = 0 from the laws of logarithms. Hence loga 1 =0. In the above graphs it is seen that log10 1 = 0 and loge 1 =0 (ii) loga a = 1 Let loga a = x, then a x = a, from the deﬁnition of a logarithm.

If a x = a then x = 1 Hence loga a = 1. (Check with a calculator that log10 10 =1 and loge e = 1.) (iii) loga 0 → −∞ Let loga 0 = x then a x = 0 from the deﬁnition of a logarithm. If a x = 0, and a is a positive real number, then x must approach minus inﬁnity. (For example, check with a calculator, 2−2 = 0.25, 2−20 = 9.54 × 10−7, 2−200 = 6.22 × 10−61 , and so on.) Hence loga 0 → −∞

This Revision test covers the material contained in Chapters 9 to 13. The marks for each question are shown in brackets at the end of each question. 1.

Solve the following pairs of simultaneous equations: (a) 7x − 3y = 23 2x + 4y = −8 b (b) 3a − 8 + = 0 8 b+

2.

3.

a 21 = 2 4

Determine the quadratic equation in x whose roots are 1 and −3 (3)

7.

Solve the equation 4x 2 − 9x + 3 = 0 correct to 3 decimal places. (5)

8.

The current i ﬂowing through an electronic device is given by: i = 0.005 v2 + 0.014 v where v is the voltage. Calculate the values of v (5) when i = 3 × 10−3

9.

for m

2(y − z) (b) x = t 1 1 1 (c) = + RT R A RB

(c)

for R A for y

p −q (e) K = 1 + pq

for q

Solve the following inequalities: (a) 2 − 5x ≤ 9 +2x (b) |3 +2t | ≤6

for z

(d) x 2 − y 2 = 3ab

(6)

6.

In an engineering process two variables x and y b are related by the equation y = ax + where a and x b are constants.Evaluate a and b if y = 15 when x = 1 and y = 13 when x = 3 (4) Transpose the following equations:

Solve the following equations by factorisation: (a) x 2 − 9 =0 (b) 2x 2 − 5x − 3 = 0

(12)

(a) y = mx + c

4.

5.

x −1 > 0 (d) (3t + 2)2 > 16 3x + 5

(e) 2x 2 − x − 3