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Answers to exercises 271 Exercise 7 (Page 12) 1. (a) 5.7% (b) 37.4% (c) 128.5% 2. (a) 21.2% (b) 79.2% (c) 169% 3. (a) 496.4 t (b) 8.657 g (c) 20.73 s 4. 2.25% 5. (a) 14% (b) 15.67% (c) 5.36% 6. 37.8 g 7. 7.2% 8. A 0.6 kg, B 0.9 kg, C 0.5 kg 9. 54%, 31%, 15%, 0.3 t 10. 20 000 kg 11. 13.5 mm, 11.5 mm 12. 600 kW Exercise 8 (Page 15) 1. (a) 3 7 (b) 4 9 2. (a) 2 6 (b) 7 10 3. (a) 2 (b) 3 5 4. (a) 5 3 (b) 7 3 5. (a) 7 6 (b) 3 6 6. (a) 15 15 (b) 17 8 7. (a) 2 (b) 3 6 8. (a) 5 2 (b) 13 2 9. (a) 3 4 (b) 1 10. (a) 5 2 (b) 1 3 5 11. (a) 7 2 (b) 1 2 Exercise 9 (Page 17) 1. (a) 1 1 (b) 3 × 5 2 7 3 × 3 7 2. (a) 32 1 (b) 2 5 2 10 × 5 2 3. (a) 9 (b) ±3 (c) ± 1 2 4. 147 148 8. −5 65 72 5. −1 19 56 Exercise 10 (Page 18) 12. (a) 1 (b) 1 (d) ± 2 3 6. −3 13 45 9. 64 10. 4 1 2 1. (a) 7.39 × 10 (b) 2.84 × 10 (c) 1.976 2 × 10 2 2. (a) 2.748 × 10 3 (b) 3.317 × 10 4 (c) 2.742 18 × 10 5 3. (a) 2.401 × 10 −1 (b) 1.74 × 10 −2 (c) 9.23 × 10 −3 4. (a) 1.702 3 × 10 3 (b) 1.004 × 10 (c) 1.09 × 10 −2 5. (a) 5 × 10 −1 (b) 1.187 5 × 10 (c) 1.306 × 10 2 (d) 3.125 × 10 −2 6. (a) 1010 (b) 932.7 (c) 54 100 (d) 7 7. (a) 0.038 9 (b) 0.674 1 (c) 0.008 7. 1 9 Exercise 11 (Page 19) 1. (a) 1.351 × 10 3 (b) 8.731 × 10 −1 (c) 5.734 × 10 −2 2. (a) 1.723 1 × 10 3 (b) 3.129 × 10 −3 (c) 1.119 9 × 10 3 3. (a) 1.35 × 10 2 (b) 1.1 × 10 −5 4. (a) 2 × 10 −2 (b) 1.5 × 10 −3 5. (a) 2.71 × 10 3 kg m −3 (b) 4.4 × 10 −1 (c) 3.767 3 × 10 2 (e) 9.578 97 × 10 7 Ckg −1 Exercise 12 (Page 20) 1. (a) 100 × 10 3 W = 100 kW (d) 5.11 × 10 −1 MeV (f ) 2.241 × 10 −2 m 3 mol −1 (b) 0.54 × 10 −3 Aor540× 10 −6 A = 0.54 mA or 540 µA (c) 1.5 × 10 6 = 1.5 M (e) 35 × 10 9 Hz = 35 GHz (g) 17 × 10 −6 A = 17 µA (d) 22.5 × 10 −3 V = 22.5 mV (f ) 15 × 10 −12 F = 15 pF (h) 46.2 × 10 3 = 46.2 k 2. (a) 25 µA (b) 1 nF (c) 620 kV (d) 1.25 M 3. (a) 13.5 × 10 −3 (b) 4 × 10 3 Exercise 13 (Page 22) 1. order of magnitude error 2. Rounding-off error – should add ‘correct to 4 significant figures’ or ‘correct to 1 decimal place’ 3. Blunder 4. Measured values, hence c = 55 800 Pa m 3 5. Order of magnitude error and rounding-off error – should be 0.022 5, correct to 3 significant figures or 0.022 5, correct to 4 decimal places 6. ≈30 (29.61, by calculator) 7. ≈2 (1.988, correct to 4 s.f., by calculator) 8. ≈10 (9.481, correct to 4 s.f., by calculator) Exercise 14 (Page 24) 1. (a) 10.56 (b) 5443 (c) 96 970 (d) 0.004 083 2. (a) 2.176 (b) 5.955 (c) 270.7 (d) 0.160 0 3. (a) 0.128 7 (b) 0.020 64 (c) 12.25 (d) 0.894 5 4. (a) 558.6 (b) 31.09 5. (a) 109.1 (b) 3.641 6. (a) 0.248 9 (b) 500.5 7. (a) 0.057 77 (b) 28.90 (c) 0.006 761
272 Basic Engineering Mathematics Exercise 18 (Page 32) 8. (a) 2 515 (b) 146.0 (c) 0.000 029 32 (c) 53.437 5 10 (d) 213.718 75 10 4. x 7/10 y 1/6 z 1/2 1 + a p 2 q 5. 6. b q − p 9. (a) 18.63 (b) 87.36 (c) 0.164 3 (d) 0.064 56 1. (a) 101 2 (b) 1111 2 (c) 10 011 2 (d) 11 101 2 10. (a) 0.005 559 (b) 1.900 2. (a) 11 111 2 (b) 101 010 2 (c) 111 001 2 (d) 111 111 2 11. (a) 6.248 (b) 0.963 0 3. (a) 0.01 2 (b) 0.001 11 2 (c) 0.010 01 2 (d) 0.10 011 2 12. (a) 1.165 (b) 2.680 4. (a) 101 111.011 01 2 (b) 11 110.110 1 2 13. (a) 1.605 (b) 11.74 (c) 110 101.111 01 2 (d) 111 101.101 01 2 14. (a) 6.874 × 10 −3 (b) 8.731 × 10 −2 Exercise 19 (Page 33) Exercise 15 (Page 25) 1. (a) 101 010 111 2 (b) 1 000 111 100 2 (c) 10 011 110 001 2 1. (a) 49.40 euro (b) $175.78 (c) £78 (d) £126 (e) $132 2. (a) 0.011 11 2 (b) 0.101 1 2 (c) 0.101 11 2 2. (a) 381 mm (b) 56.35 km/h (c) 378.35 km 3. (a) 11 110 111.000 11 2 (b) 1 000 000 010.011 1 2 (d) 52 lb 13 oz (e) 6.82 kg (f) 54.55 litres (g) 5.5 gallons (c) 11 010 110 100.110 01 2 3. (a) 7.09 a.m. (b) 51 minutes, 32 m.p.h. (c) 7.04 a.m. 4. (a) 7.437 5 10 (b) 41.25 10 (c) 7 386.187 5 10 Exercise 16 (Page 28) Exercise 20 (Page 36) 1. A = 66.59 cm 2 2. C = 52.78 mm 1. 231 10 2. 44 10 3. 152 10 4. 753 10 3. R = 37.5 4. 159 m/s 5. 36 16 6. C8 16 7. 5B 16 8. EE 16 5. 0.407A 6. 0.005 02 m or 5.02 mm 9. D7 16 10. EA 16 11. 8B 16 12. A5 16 7. 0.144 J 8. 628.8 m 2 13. 110 111 2 14. 11 101 101 2 15. 10 011 111 2 9. 224.5 10. 14 230 kg/m 3 16. 101 000 100 001 2 11. 281.1 m/s 12. 2.526 13. £589.27 14. 508.1 W Exercise 21 (Page 39) 15. V = 2.61V 16. F = 854.5 1. −16 2. −8 3. 4a 17. I = 3.81A 18. T = 14.79 s 4. 2 1 6 c 5. −2 1 4 x + 1 1 2 y + 2z 19. E = 3.96 J 20. I = 12.77A 6. a − 4b − c 7. 9d − 2e 8. 3x − 5y + 5z 21. S = 17.25 m 22. A = 7.184 cm 2 23. v = 7.327 9. −5 1 2 a + 5 6 b − 4c 10. 3x2 − xy − 2y 2 Exercise 17 (Page 30) 11. 6a 2 − 13ab + 3ac − 5b 2 + bc 1. (a) 6 10 (b) 11 10 (c) 14 10 (d) 9 10 2. (a) 21 10 (b) 25 10 (c) 45 10 (d) 51 10 12. (i) 1 3b (ii) 2ab 13. 2x − y 14. 3p + 2q 3. (a) 0.812 5 10 (b) 0.781 25 10 Exercise 22 (Page 41) (c) 0.218 75 10 (d) 0.343 75 10 4. (a) 26.75 10 (b) 23.375 10 1. x 5 y 4 z 3 ,13 1 2 2. a 2 b 1/2 c −2 ,4 1 2 3. a 3 b −2 c,9
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Basic Engineering Mathematics
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Basic Engineering Mathematics Fourt
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Contents Preface xi 1. Basic arithm
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Contents vii 13.3 Graphical solutio
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Contents ix 27.4 Vector subtraction
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Preface Basic Engineering Mathemati
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1 Basic arithmetic 1.1 Arithmetic o
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Basic arithmetic 3 12. −23148 −
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Basic arithmetic 5 Problem 18. 23
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Fractions, decimals and percentages
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Indices and standard form 15 From l
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Indices and standard form 17 16 2
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Indices and standard form 19 3.6 Fu
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4 Calculations and evaluation of fo
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Calculations and evaluation of form
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Computer numbering systems 31 3. (a
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Computer numbering systems 33 11 11
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Computer numbering systems 35 Probl
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6 Algebra 6.1 Basic operations Alge
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Algebra 39 ) 2a 2 − 2ab − b 2 2
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Algebra 41 Using the third and four
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Algebra 43 x 2 is a common factor o
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Algebra 45 10. p 2 − 3pq × 2p ÷
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7 Simple equations 7.1 Expressions,
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Simple equations 49 7.3 Further wor
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Simple equations 51 Problem 18. A r
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Simple equations 53 Squaring both s
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Transposition of formulae 55 Rearra
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Transposition of formulae 57 Now tr
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Transposition of formulae 59 6. 7.
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Simultaneous equations 61 Hence y =
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Simultaneous equations 63 It is oft
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Simultaneous equations 65 Thus the
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Simultaneous equations 67 Substitut
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10 Quadratic equations 10.1 Introdu
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Quadratic equations 71 In Problems
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Quadratic equations 73 Summarizing:
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Quadratic equations 75 Neglecting t
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11 Inequalities 11.1 Introduction t
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Inequalities 79 i.e. 11.4 Inequalit
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Inequalities 81 Solving quadratic i
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12 Straight line graphs 12.1 Introd
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Straight line graphs 85 Problem 2.
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Straight line graphs 87 (b) Rearran
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Straight line graphs 89 Degrees Fah
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Straight line graphs 91 y 147 140 A
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Straight line graphs 93 Stress (pas
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Graphical solution of equations 95
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14 Logarithms 14.1 Introduction to
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Logarithms 105 i.e. −5 = 4x i.e.
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15 Exponential functions 15.1 The e
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Exponential functions 109 If in the
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Exponential functions 111 Problem 1
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Exponential functions 113 ( ) 5.14
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Exponential functions 115 (b) Trans
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16 Reduction of non-linear laws to
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Reduction of non-linear laws to lin
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Graphs with logarithmic scales 125
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18 Geometry and triangles 18.1 Angu
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Geometry and triangles 133 (d) 227
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Geometry and triangles 135 (a) Equi
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Geometry and triangles 137 Hence XZ
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Geometry and triangles 139 Now try
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Geometry and triangles 141 Assignme
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Introduction to trigonometry 143 Fr
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Introduction to trigonometry 145 Si
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Introduction to trigonometry 147 If
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Introduction to trigonometry 149 To
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20 Trigonometric waveforms 20.1 Gra
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Trigonometric waveforms 153 between
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Trigonometric waveforms 155 45° 60
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Trigonometric waveforms 157 y 4 0 y
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Trigonometric waveforms 159 v rads/
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Trigonometric waveforms 161 Assignm
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Cartesian and polar co-ordinates 16
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Cartesian and polar co-ordinates 16
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Areas of plane figures 167 W X Tabl
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Areas of plane figures 169 (b) Area
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Areas of plane figures 171 14. Calc
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Areas of plane figures 173 is 12 50
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The circle 175 Q B Now try the foll
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The circle 177 From equation (2), a
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The circle 179 Thus, for example, t
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Volumes of common solids 181 Proble
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Volumes of common solids 183 Proble
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Volumes of common solids 185 Fig. 2
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Volumes of common solids 187 From F
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Volumes of common solids 189 Volume
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25 Irregular areas and volumes and
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Irregular areas and volumes and mea
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Triangles and some practical applic
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27 Vectors 27.1 Introduction Some p
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Vectors 209 r 10° b Having obtaine
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Vectors 211 b Fig. 27.11 o Fig. 27.
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Vectors 213 N Thus the velocity of
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Adding of waveforms 215 y 6.1 6 4 2
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Adding of waveforms 217 The horizon
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Number sequences 219 The first four
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Page 236 and 237: Number sequences 223 Hence 1 ( 1 9
Page 238 and 239: Number sequences 225 Now try the fo
Page 240 and 241: Presentation of statistical data 22
Page 248 and 249: 31 Measures of central tendency and
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Page 254 and 255: 32 Probability 32.1 Introduction to
Page 256 and 257: Probability 243 (a) The probability
Page 258 and 259: Probability 245 Two brass washers a
Page 260 and 261: 33 Introduction to differentiation
Page 262 and 263: Introduction to differentiation 249
Page 270 and 271: 34 Introduction to integration 34.1
Page 272 and 273: Introduction to integration 259 ∫
Page 274 and 275: Introduction to integration 261 Pro
Page 276 and 277: Introduction to integration 263 A t
Page 278 and 279: Introduction to integration 265 Ass
Page 280 and 281: List of formulae 267 (ii) Parallelo
Page 282 and 283: List of formulae 269 Cartesian and
Page 286 and 287: Answers to exercises 273 7. ab 6 c
Page 288 and 289: Answers to exercises 275 5. 0.013 3
Page 290 and 291: Answers to exercises 277 Exercise 5
Page 298 and 299: Index Abscissa, 83 Acute angle, 132
Page 300 and 301: Index 287 Interior angles, 132, 134
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