Bird’s Basic Engineering Mathematics Whyisknowledgeofmathematicsimportantinengineering? A career in any engineering or scientific field will concerned with release, control and utilisation of requirebothbasicandadvancedmathematics.Without nuclearenergyandnuclearwastedisposal. mathematics to determine principles, calculate dimen- Petroleum engineers require mathematics to devise sionsandlimits,explorevariations,proveconcepts,and methods to improve oil and gas well production and soon,therewouldbenomobiletelephones,televisions, determine the need for new or modified tool designs; stereo systems, video games, microwave ovens, com- they oversee drilling and offer technical advice to putersorvirtuallyanythingelectronic.Therewouldbe achieveeconomicalandsatisfactoryprogress. no bridges, tunnels, roads, skyscrapers, automobiles, ships,planes,rocketsormostthingsmechanical.There Industrial engineers require mathematics to design, wouldbenometalsbeyondthecommonones,suchas develop,testandevaluateintegratedsystemsforman- ironandcopper,noplastics,nosynthetics.Infact,soci- agingindustrialproductionprocesses,includinghuman etywouldmostcertainlybelessadvancedwithoutthe work factors, quality control, inventory control, logis- use of mathematics throughout the centuries and into tics and material flow, cost analysis and production thefuture. coordination. Electrical engineers require mathematics to design, Environmental engineers require mathematics to develop,test,orsupervisethemanufacturingandinstal- design, plan, or perform engineering duties in the lation of electrical equipment, components or systems prevention, control and remediation of environmen- forcommercial,industrial,militaryorscientificuse. tal health hazards, using various engineering disci- Mechanicalengineersrequiremathematicstoperform plines; their work may include waste treatment, site engineering duties in planning and designing tools, remediationorpollutioncontroltechnology. engines,machinesandothermechanicallyfunctioning Civil engineers require mathematics in all levels in equipment;theyoverseeinstallation,operation,mainte- civil engineering – structural engineering, hydraulics nanceandrepairofsuchequipmentascentralisedheat, andgeotechnicalengineeringareallfieldsthatemploy gas,waterandsteamsystems. mathematical tools such as differential equations, ten- Aerospace engineers require mathematics to perform soranalysis,fieldtheory,numericalmethodsandoper- a variety of engineering work in designing, construct- ationsresearch. ing and testing aircraft, missiles, and spacecraft; they conduct basic and applied research to evaluate adapt- Knowledgeofmathematicsisthereforeneededbyeach abilityofmaterialsandequipmenttoaircraftdesignand oftheengineeringdisciplineslistedabove. manufacture and recommend improvements in testing Itisintendedthatthistext– Bird’sBasicEngineering equipmentandtechniques. Mathematics–willprovideastepbystepapproachto Nuclear engineers require mathematics to conduct learningalltheearly,fundamentalmathematicsneeded research on nuclear engineering problems or apply foryourfutureengineeringstudies. principles and theory of nuclear science to problems Now in its eighth edition, Bird’s Basic Engineer- John Bird, BSc (Hons), CEng, CMath, CSci, FIMA, ing Mathematics has helped thousands of students FIET,FCollT,istheformerHeadofAppliedElectron- to succeed in their exams. Mathematical theories ics in the Faculty of Technology at Highbury College, are explained in a straightforward manner, supported Portsmouth,UK.Morerecently,hehascombinedfree- by practical engineering examples and applications lance lecturing at the University of Portsmouth, with to ensure that readers can relate theory to practice. Examiner responsibilities for Advanced Mathematics Some1,000engineeringsituations/problemshavebeen with City and Guilds and examining for the Inter- ‘flagged-up’tohelpdemonstratethatengineeringcan- national Baccalaureate Organisation. He has over 45 not be fully understood without a good knowledge of years’ experience of successfully teaching, lecturing, mathematics. instructing, training, educating and planning trainee The extensive and thorough coverage makes this a engineers study programmes. He is the author of 146 great text for introductory level engineering courses – textbooks on engineering, science and mathematical such as for aeronautical, construction, electrical, elec- subjects, with worldwide sales of over one million tronic, mechanical, manufacturing engineering and copies. He is a chartered engineer, a chartered math- vehicle technology – including for BTEC First, ematician, a chartered scientist and a Fellow of three NationalandDiplomasyllabuses,City&GuildsTech- professional institutions. He has recently retired from nicianCertificateandDiplomasyllabuses,andevenfor lecturing at the Defence College of Marine Engineer- GCSErevision. ing in the Defence College of Technical Training at Its companion website provides extra materials for H.M.S. Sultan, Gosport, Hampshire, UK, one of the students and lecturers, including full solutions for all largestengineeringtrainingestablishmentsinEurope. 1,700 further questions, lists of essential formulae, multiple choice tests, and illustrations, as well as full solutionstorevisiontestsforcourseinstructors. To Sue Bird’s Basic Engineering Mathematics Eighth Edition John Bird Eightheditionpublished2021 byRoutledge 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN andbyRoutledge 52VanderbiltAvenue,NewYork,NY10017 RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2021JohnBird TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78of theCopyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor retrievalsystem,withoutpermissioninwritingfromthepublishers. Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification andexplanationwithoutintenttoinfringe. FirsteditionpublishedbyNewnes1999 SeventheditionpublishedbyRoutledge2017 BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Acatalogrecordhasbeenrequestedforthisbook ISBN:9780367643706(hbk) ISBN:9780367643676(pbk) ISBN:9781003124214(ebk) TypesetinTimesby CenveoPublisherServices Visitthecompanionwebsite:www.routledge.com/cw/bird Contents Preface xi 6.3 Directproportion 50 6.4 Inverseproportion 54 Acknowledgements xiii 1 Basicarithmetic 1 7 Powers,rootsandlawsofindices 57 1.1 Introduction 1 7.1 Introduction 57 1.2 Revisionofadditionandsubtraction 2 7.2 Powersandroots 57 1.3 Revisionofmultiplicationanddivision 3 7.3 Lawsofindices 59 1.4 Highestcommonfactorsandlowest commonmultiples 5 1.5 Orderofoperationandbrackets 7 8 Units,prefixesandengineeringnotation 64 8.1 Introduction 64 2 Fractions 10 8.2 SIunits 64 2.1 Introduction 10 8.3 Commonprefixes 65 2.2 Addingandsubtractingfractions 11 8.4 Standardform 68 2.3 Multiplicationanddivisionoffractions 13 8.5 Engineeringnotation 70 2.4 Orderofoperationwithfractions 15 8.6 Metricconversions 72 8.7 Metric-US/Imperialconversions 76 RevisionTest1 18 RevisionTest3 82 3 Decimals 19 3.1 Introduction 19 3.2 Convertingdecimalstofractionsand 9 Basicalgebra 83 viceversa 19 9.1 Introduction 83 3.3 Significantfiguresanddecimalplaces 21 9.2 Basicoperations 84 3.4 Addingandsubtractingdecimal 9.3 Lawsofindices 87 numbers 22 3.5 Multiplyinganddividingdecimal 10 Furtheralgebra 91 numbers 23 10.1 Introduction 91 10.2 Brackets 91 4 Usingacalculator 26 10.3 Factorisation 93 4.1 Introduction 26 10.4 Lawsofprecedence 94 4.2 Adding,subtracting,multiplyingand dividing 26 4.3 Furthercalculatorfunctions 28 11 Solvingsimpleequations 97 4.4 Evaluationofformulae 32 11.1 Introduction 97 11.2 Solvingequations 97 5 Percentages 38 11.3 Practicalproblemsinvolvingsimple 5.1 Introduction 38 equations 101 5.2 Percentagecalculations 39 5.3 Furtherpercentagecalculations 40 RevisionTest4 107 5.4 Morepercentagecalculations 42 RevisionTest2 46 12 Transposingformulae 108 12.1 Introduction 108 6 Ratioandproportion 47 12.2 Transposingformulae 108 6.1 Introduction 47 12.3 Furthertransposingofformulae 110 6.2 Ratios 48 12.4 Moredifficulttransposingofformulae 113 viii Contents 13 Solvingsimultaneousequations 118 18 Graphsreducingnon-linearlawstolinearform 185 13.1 Introduction 118 18.1 Introduction 185 13.2 Solvingsimultaneousequationsintwo 18.2 Determinationoflaw 185 unknowns 118 18.3 Revisionoflawsoflogarithms 188 13.3 Furthersolvingofsimultaneousequations 120 18.4 Determinationoflawsinvolving 13.4 Solvingmoredifficultsimultaneous logarithms 189 equations 122 19 Graphicalsolutionofequations 194 13.5 Practicalproblemsinvolving 19.1 Graphicalsolutionofsimultaneous simultaneousequations 124 equations 194 13.6 Solvingsimultaneousequationsinthree 19.2 Graphicalsolutionofquadraticequations 196 unknowns 128 19.3 Graphicalsolutionoflinearand quadraticequationssimultaneously 200 RevisionTest5 131 19.4 Graphicalsolutionofcubic equations 200 20 Graphswithlogarithmicscales 203 14 Solvingquadraticequations 132 20.1 Logarithmicscalesandlogarithmic 14.1 Introduction 132 graphpaper 203 14.2 Solutionofquadraticequationsby 20.2 Graphsoftheformy=axn 204 factorisation 133 14.3 Solutionofquadraticequationsby 20.3 Graphsoftheformy=abx 207 ‘completingthesquare’ 135 20.4 Graphsoftheformy=aekx 208 14.4 Solutionofquadraticequationsby formula 137 RevisionTest7 211 14.5 Practicalproblemsinvolvingquadratic equations 138 21 Anglesandtriangles 213 14.6 Solutionoflinearandquadratic 21.1 Introduction 213 equationssimultaneously 141 21.2 Angularmeasurement 213 21.3 Triangles 219 15 Logarithms 143 21.4 Congruenttriangles 223 15.1 Introductiontologarithms 143 21.5 Similartriangles 225 15.2 Lawsoflogarithms 145 21.6 Constructionoftriangles 227 15.3 Indicialequations 147 22 Introductiontotrigonometry 230 15.4 Graphsoflogarithmicfunctions 149 22.1 Introduction 230 22.2 ThetheoremofPythagoras 230 16 Exponentialfunctions 151 22.3 Sines,cosinesandtangents 233 16.1 Introductiontoexponentialfunctions 151 22.4 Evaluatingtrigonometricratiosofacute 16.2 Thepowerseriesforex 152 angles 235 16.3 Graphsofexponentialfunctions 154 22.5 Solvingright-angledtriangles 238 16.4 Napierianlogarithms 156 22.6 Anglesofelevationanddepression 241 16.5 Lawsofgrowthanddecay 159 RevisionTest8 245 RevisionTest6 164 23 Trigonometricwaveforms 247 23.1 Graphsoftrigonometricfunctions 247 23.2 Anglesofanymagnitude 248 17 Straightlinegraphs 165 23.3 Theproductionofsineandcosinewaves 251 17.1 Introductiontographs 165 23.4 Terminologyinvolvedwithsineand 17.2 Axes,scalesandco-ordinates 165 cosinewaves 251 17.3 Straightlinegraphs 167 23.5 Sinusoidalform:Asin(!t(cid:6)(cid:11)) 254 17.4 Gradients,interceptsandequations ofgraphs 170 17.5 Practicalproblemsinvolvingstraight linegraphs 177 Contents ix 24 Non-right-angledtrianglesandsomepractical 29 Irregularareasandvolumesandmeanvalues 318 applications 258 29.1 Areasofirregularfigures 318 24.1 Thesineandcosinerules 258 29.2 Volumesofirregularsolids 321 24.2 Areaofanytriangle 259 29.3 Meanoraveragevaluesofwaveforms 322 24.3 Workedproblemsonthesolutionof trianglesandtheirareas 259 RevisionTest11 327 24.4 Furtherworkedproblemsonthesolution oftrianglesandtheirareas 261 30 Vectors 329 24.5 Practicalsituationsinvolving 30.1 Introduction 329 trigonometry 262 30.2 Scalarsandvectors 329 24.6 Furtherpracticalsituationsinvolving 30.3 Drawingavector 330 trigonometry 264 30.4 Additionofvectorsbydrawing 331 30.5 Resolvingvectorsintohorizontaland 25 Cartesianandpolarco-ordinates 268 verticalcomponents 333 25.1 Introduction 268 30.6 Additionofvectorsbycalculation 334 25.2 ChangingfromCartesiantopolar 30.7 Vectorsubtraction 338 co-ordinates 268 30.8 Relativevelocity 339 25.3 ChangingfrompolartoCartesian 30.9 i,jandknotation 340 co-ordinates 270 25.4 UseofPol/Recfunctionson calculators 271 31 Methodsofaddingalternating waveforms 343 31.1 Combiningtwoperiodicfunctions 343 RevisionTest9 273 31.2 Plottingperiodicfunctions 344 31.3 Determiningresultantphasorsby drawing 345 26 Areasofcommonshapes 274 31.4 Determiningresultantphasorsbythe 26.1 Introduction 274 sineandcosinerules 347 26.2 Commonshapes 274 31.5 Determiningresultantphasorsby 26.3 Areasofcommonshapes 277 horizontalandverticalcomponents 348 26.4 Areasofsimilarshapes 285 RevisionTest12 352 27 Thecircleanditsproperties 287 27.1 Introduction 287 27.2 Propertiesofcircles 287 32 Presentationofstatisticaldata 354 27.3 Radiansanddegrees 289 32.1 Somestatisticalterminology 355 27.4 Arclengthandareaofcirclesand 32.2 Presentationofungroupeddata 356 sectors 290 32.3 Presentationofgroupeddata 359 27.5 Theequationofacircle 294 33 Mean,median,modeandstandarddeviation 367 33.1 Measuresofcentraltendency 367 RevisionTest10 297 33.2 Mean,medianandmodefordiscrete data 368 28 Volumesandsurfaceareasofcommonsolids 299 33.3 Mean,medianandmodeforgrouped 28.1 Introduction 299 data 369 28.2 Volumesandsurfaceareasofcommon 33.4 Standarddeviation 370 shapes 299 33.5 Quartiles,decilesandpercentiles 372 28.3 Summaryofvolumesandsurfaceareas ofcommonsolids 306 34 Probability 375 28.4 Morecomplexvolumesandsurface 34.1 Introductiontoprobability 376 areas 306 34.2 Lawsofprobability 377 28.5 Volumesandsurfaceareasoffrustaof pyramidsandcones 312 RevisionTest13 384 28.6 Volumesofsimilarshapes 316