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Basic Engineering Mathematics In memory of Elizabeth Basic Engineering Mathematics Fifth edition John Bird ,BSc(Hons), CMath, CEng, CSci, FIMA,FIET, MIEE, FIIE, FCollT AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO NewnesisanimprintofElsevier NewnesisanimprintofElsevier TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK 30CorporateDrive,Suite400,Burlington,MA01803,USA Firstedition 1999 Secondedition 2000 Thirdedition 2002 Fourthedition 2005 Fifthedition 2010 Copyright©2010,JohnBird,PublishedbyElsevierLtd.Allrightsreserved. TherightofJohnBirdtobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith theCopyright,DesignsandPatentsAct1988. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyform orbyanymeanselectronic,mechanical,photocopying,recordingorotherwisewithoutthepriorwritten permissionofthepublisher. 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ISBN-13:978-1-85-617697-2 ForinformationonallNewnespublications visitourWebsiteatwww.elsevierdirect.com Typesetby:diacriTech,India PrintedandboundinChina 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 Contents Preface ix 6.3 Directproportion 42 Acknowledgements x 6.4 Inverseproportion 45 Instructor’sManual xi 7 Powers,rootsandlawsofindices 47 7.1 Introduction 47 1 Basicarithmetic 1 7.2 Powersandroots 47 1.1 Introduction 1 7.3 Lawsofindices 48 1.2 Revisionofadditionandsubtraction 1 1.3 Revisionofmultiplicationanddivision 3 8 Units,prefixesandengineeringnotation 53 1.4 Highestcommonfactorsandlowest 8.1 Introduction 53 commonmultiples 5 8.2 SIunits 53 1.5 Orderofprecedenceandbrackets 6 8.3 Commonprefixes 53 8.4 Standardform 56 2 Fractions 9 8.5 Engineeringnotation 57 2.1 Introduction 9 2.2 Addingandsubtractingfractions 10 RevisionTest3 60 2.3 Multiplicationanddivisionoffractions 12 2.4 Orderofprecedencewithfractions 13 9 Basicalgebra 61 9.1 Introduction 61 RevisionTest1 15 9.2 Basicoperations 61 9.3 Lawsofindices 64 3 Decimals 16 3.1 Introduction 16 10 Furtheralgebra 68 3.2 Convertingdecimalstofractionsand 10.1 Introduction 68 vice-versa 16 10.2 Brackets 68 3.3 Significantfiguresanddecimalplaces 17 10.3 Factorization 69 3.4 Addingandsubtractingdecimalnumbers 18 10.4 Lawsofprecedence 71 3.5 Multiplyinganddividingdecimalnumbers 19 11 Solvingsimpleequations 73 4 Usingacalculator 22 11.1 Introduction 73 4.1 Introduction 22 11.2 Solvingequations 73 4.2 Adding,subtracting,multiplyingand 11.3 Practicalproblemsinvolvingsimple dividing 22 equations 77 4.3 Furthercalculatorfunctions 23 4.4 Evaluationofformulae 28 RevisionTest4 82 5 Percentages 33 12 Transposingformulae 83 5.1 Introduction 33 12.1 Introduction 83 5.2 Percentagecalculations 33 12.2 Transposingformulae 83 5.3 Furtherpercentagecalculations 35 12.3 Furthertransposingofformulae 85 5.4 Morepercentagecalculations 36 12.4 Moredifficulttransposingofformulae 87 RevisionTest2 39 13 Solvingsimultaneousequations 90 13.1 Introduction 90 6 Ratioandproportion 40 13.2 Solvingsimultaneousequationsintwo 6.1 Introduction 40 unknowns 90 6.2 Ratios 40 13.3 Furthersolvingofsimultaneousequations 92 vi Contents 13.4 Solvingmoredifficultsimultaneous 19.2 Graphicalsolutionofquadraticequations 156 equations 94 19.3 Graphicalsolutionoflinearandquadratic 13.5 Practicalproblemsinvolvingsimultaneous equationssimultaneously 160 equations 96 19.4 Graphicalsolutionofcubicequations 161 13.6 Solvingsimultaneousequationsinthree unknowns 99 RevisionTest7 163 RevisionTest5 101 20 Anglesandtriangles 165 20.1 Introduction 165 14 Solvingquadraticequations 102 20.2 Angularmeasurement 165 14.1 Introduction 102 20.3 Triangles 171 14.2 Solutionofquadraticequationsby 20.4 Congruenttriangles 175 factorization 102 20.5 Similartriangles 176 14.3 Solutionofquadraticequationsby 20.6 Constructionoftriangles 179 ‘completingthesquare’ 105 14.4 Solutionofquadraticequationsby 21 Introductiontotrigonometry 181 formula 106 21.1 Introduction 181 14.5 Practicalproblemsinvolvingquadratic 21.2 ThetheoremofPythagoras 181 equations 108 21.3 Sines,cosinesandtangents 183 14.6 Solutionoflinearandquadraticequations 21.4 Evaluatingtrigonometricratiosofacute simultaneously 110 angles 185 15 Logarithms 111 21.5 Solvingright-angledtriangles 188 15.1 Introductiontologarithms 111 21.6 Anglesofelevationanddepression 191 15.2 Lawsoflogarithms 113 15.3 Indicialequations 115 RevisionTest8 193 15.4 Graphsoflogarithmicfunctions 116 16 Exponentialfunctions 118 22 Trigonometricwaveforms 195 16.1 Introductiontoexponentialfunctions 118 22.1 Graphsoftrigonometricfunctions 195 16.2 Thepowerseriesforex 119 22.2 Anglesofanymagnitude 196 16.3 Graphsofexponentialfunctions 120 22.3 Theproductionofsineandcosinewaves 198 16.4 Napierianlogarithms 122 22.4 Terminologyinvolvedwithsineand 16.5 Lawsofgrowthanddecay 125 cosinewaves 199 22.5 Sinusoidalform: Asin(ωt±α) 202 RevisionTest6 129 23 Non-right-angledtrianglesandsomepractical applications 205 17 Straightlinegraphs 130 23.1 Thesineandcosinerules 205 17.1 Introductiontographs 130 23.2 Areaofanytriangle 205 17.2 Axes,scalesandco-ordinates 130 23.3 Workedproblemsonthesolutionof 17.3 Straightlinegraphs 132 trianglesandtheirareas 206 17.4 Gradients,interceptsandequations 23.4 Furtherworkedproblemsonthesolution ofgraphs 134 oftrianglesandtheirareas 207 17.5 Practicalproblemsinvolvingstraightline 23.5 Practicalsituationsinvolvingtrigonometry 209 graphs 141 23.6 Furtherpracticalsituationsinvolving 18 Graphsreducingnon-linearlawstolinearform 147 trigonometry 211 18.1 Introduction 147 24 Cartesianandpolarco-ordinates 214 18.2 Determinationoflaw 147 24.1 Introduction 214 18.3 Revisionoflawsoflogarithms 150 24.2 ChangingfromCartesiantopolar 18.4 Determinationoflawinvolvinglogarithms 150 co-ordinates 214 19 Graphicalsolutionofequations 155 24.3 ChangingfrompolartoCartesian 19.1 Graphicalsolutionofsimultaneous co-ordinates 216 equations 155 24.4 UseofPol/Recfunctionsoncalculators 217 vii Contents RevisionTest9 218 30.4 Determiningresultantphasorsbythesine andcosinerules 281 30.5 Determiningresultantphasorsby 25 Areasofcommonshapes 219 horizontalandverticalcomponents 283 25.1 Introduction 219 25.2 Commonshapes 219 RevisionTest12 286 25.3 Areasofcommonshapes 221 25.4 Areasofsimilarshapes 229 31 Presentationofstatisticaldata 288 26 Thecircle 230 31.1 Somestatisticalterminology 288 26.1 Introduction 230 31.2 Presentationofungroupeddata 289 26.2 Propertiesofcircles 230 31.3 Presentationofgroupeddata 292 26.3 Radiansanddegrees 232 26.4 Arclengthandareaofcirclesandsectors 233 32 Mean,median,modeandstandarddeviation 299 26.5 Theequationofacircle 236 32.1 Measuresofcentraltendency 299 32.2 Mean,medianandmodefordiscretedata 299 32.3 Mean,medianandmodeforgroupeddata 300 RevisionTest10 238 32.4 Standarddeviation 302 32.5 Quartiles,decilesandpercentiles 303 27 Volumesofcommonsolids 240 27.1 Introduction 240 33 Probability 306 27.2 Volumesandsurfaceareasofcommon 33.1 Introductiontoprobability 306 shapes 240 33.2 Lawsofprobability 307 27.3 Summaryofvolumesandsurfaceareasof commonsolids 247 RevisionTest13 312 27.4 Morecomplexvolumesandsurfaceareas 247 27.5 Volumesandsurfaceareasoffrustaof pyramidsandcones 252 34 Introductiontodifferentiation 313 34.1 Introductiontocalculus 313 27.6 Volumesofsimilarshapes 256 34.2 Functionalnotation 313 28 Irregularareasandvolumes,andmeanvalues 257 34.3 Thegradientofacurve 314 28.1 Areasofirregularfigures 257 34.4 Differentiationfromfirstprinciples 315 28.2 Volumesofirregularsolids 259 34.5 Differentiationofy=axn bythe 28.3 Meanoraveragevaluesofwaveforms 260 generalrule 315 34.6 Differentiationofsineandcosinefunctions 318 34.7 Differentiationofeaxandlnax 320 RevisionTest11 264 34.8 Summaryofstandardderivatives 321 34.9 Successivedifferentiation 322 29 Vectors 266 34.10 Ratesofchange 323 29.1 Introduction 266 29.2 Scalarsandvectors 266 35 Introductiontointegration 325 29.3 Drawingavector 266 35.1 Theprocessofintegration 325 29.4 Additionofvectorsbydrawing 267 35.2 Thegeneralsolutionofintegralsofthe 29.5 Resolvingvectorsintohorizontaland formaxn 325 verticalcomponents 269 35.3 Standardintegrals 326 29.6 Additionofvectorsbycalculation 270 35.4 Definiteintegrals 328 29.7 Vectorsubtraction 274 35.5 Theareaunderacurve 330 29.8 Relativevelocity 276 29.9 i, j andknotation 277 RevisionTest14 335 30 Methodsofaddingalternatingwaveforms 278 Listofformulae 336 30.1 Combiningtwoperiodicfunctions 278 30.2 Plottingperiodicfunctions 278 Answerstopracticeexercises 340 30.3 Determiningresultantphasorsbydrawing 280 Index 356 viii Contents Website Chapters (Goto:http://www.booksite.elsevier.com/newnes/bird) Preface iv 38.3 Inequalitiesinvolvingamodulus 20 36 Numbersequences 1 38.4 Inequalitiesinvolvingquotients 21 36.1 Simplesequences 1 38.5 Inequalitiesinvolvingsquarefunctions 22 36.2 Then’thtermofaseries 1 38.6 Quadraticinequalities 23 36.3 Arithmeticprogressions 2 39 Graphswithlogarithmicscales 25 36.4 Geometricprogressions 5 39.1 Logarithmicscalesandlogarithmic 37 Binary,octalandhexadecimal 9 graphpaper 25 37.1 Introduction 9 39.2 Graphsoftheformy=axn 25 37.2 Binarynumbers 9 39.3 Graphsoftheformy=abx 28 37.3 Octalnumbers 12 39.4 Graphsoftheformy=aekx 29 37.4 Hexadecimalnumbers 15 38 Inequalities 19 RevisionTest15 32 38.1 Introductiontoinequalities 19 38.2 Simpleinequalities 19 Answerstopracticeexercises 33 Preface Basic Engineering Mathematics 5th Edition intro- asproblemsolvingisextensivelyusedtoestablishand ducesandthenconsolidatesbasicmathematicalprinci- exemplifythetheory.Itisintendedthatreaderswillgain plesandpromotesawarenessofmathematicalconcepts realunderstandingthroughseeingproblemssolvedand forstudentsneedingabroadbaseforfurthervocational thensolvingsimilarproblemsthemselves. studies. This textbook contains some 750 worked problems, Inthisfifthedition,newmaterialhasbeenaddedtomany followed by over 1550 further problems (all with of the chapters, particularly some of the earlier chap- answersattheendofthebook)containedwithinsome ters,togetherwithextrapracticalproblemsinterspersed 161 Practice Exercises; each Practice Exercise fol- throughout the text. The extent of this fifth edition lows on directlyfrom the relevant section of work. In is such that four chapters from the previous edition addition,376linediagramsenhanceunderstandingof have been removed and placed on the easily available the theory. Where at all possible,the problemsmirror website http://www.booksite.elsevier.com/newnes/bird. potentialpractical situationsfoundinengineering and The chapters removed to the website are ‘Number science. sequences’,‘Binary,octalandhexadecimal’,‘Inequali- Placed at regular intervals throughout the text are ties’and‘Graphswithlogarithmicscales’. 14 Revision Tests (plus another for the website Thetextisrelevantto: chapters) to check understanding. For example, Revi- sion Test 1 covers material contained in Chapters 1 • ‘Mathematics for Engineering Technicians’ for and 2, Revision Test 2 covers the material contained BTEC FirstNQF Level 2 –Chapters1–12,16–18, in Chapters 3–5, and so on. These Revision Tests do 20,21,23and25–27areneededforthismodule. not have answers given since it is envisaged that lec- • Themandatory‘MathematicsforTechnicians’for turers/instructors could set the tests for students to BTECNationalCertificateandNationalDiplomain attempt as part of their course structure. Lecturers/in- Engineering,NQFLevel3–Chapters7–10,14–17, structors may obtain a complimentary set of solu- 19, 20–23, 25–27, 31, 32, 34 and 35 are needed tionsoftheRevisionTestsinan Instructor’sManual, and,inaddition,Chapters1–6,11and12arehelpful available from the publishers via the internet – see revisionforthismodule. http://www.booksite.elsevier.com/newnes/bird. • Basic mathematics for a wide range of introduc- Attheendofthebookalistofrelevantformulaecon- tory/access/foundationmathematicscourses. tained within the text is included for convenience of • GCSErevisionandforsimilarmathematicscourses reference. inEnglish-speakingcountriesworldwide. Theprincipleof learningbyexampleisattheheartof BasicEngineeringMathematics5thEditionprovidesa BasicEngineeringMathematics5thEdition. leadintoEngineeringMathematics6thEdition. Each topic considered in the text is presented in a JOHNBIRD way that assumes in the reader little previous know- RoyalNavalSchoolofMarineEngineering ledge of that topic. Each chapter begins with a brief HMSSultan,formerlyUniversityofPortsmouth outlineof essential theory, definitions, formulae, laws andHighburyCollege,Portsmouth andprocedures;however,thesearekepttoaminimum