Answers to practice exercises | 57 | v5 | Basic Engineering Mathematic

ABSTRACT

L = XL 2π f

11. R = E I

12. x = a(y − 3)

13. C = 5 9 (F − 32) 14. f = 1

1. r = S − a S

or 1− a S

2. x = d λ (y +λ) or d + yd

3. f = 3F − AL 3

or f = F − AL 3

4. D = AB 2

5Ey 5. t = R − R0 R0α

6. R2 = RR1R1 − R

7. R = E − e − Ir I

or R = E − e I

− r

8. b = √( y

4ac2 )

9. x = ay√ (y2 − b2)

10. L = t 2g

4π2 11. u = √v2 − 2as

12. R = √(

360A πθ

) 13. a = N2y − x

14. L = √

Z 2 − R2 2π f ,0.080

1. a = √(

x y m − n

) 2. R = 4

√( M π

+ r4 )

3. r = 3(x + y) (1− x − y) 4. L =

mrCR μ−m

5. b = c√ 1− a2 6. r =

√( x − y x + y

)

7. b = a(p 2 − q2)

2(p2 + q2) 8. v = u f

u − f ,30

9. t2 = t1 + Q mc

,55 10. v = √(

2dgh 0.03L

) ,0.965

11. L = 8S 2

+ d,2.725

12. C =

13. 64mm 14. λ = 5 √(

aμ

ρCZ 4n

1. x = 4, y = 2 2. x = 3, y = 4 3. x = 2, y = 1.5 4. x = 4, y = 1 5. p = 2,q = −1 6. x = 1, y = 2 7. x = 3, y = 2 8. a = 2,b = 3 9. a = 5,b = 2 10. x = 1, y = 1

11. s = 2, t = 3 12. x = 3, y = −2 13. m = 2.5,n = 0.5 14. a = 6,b = −1 15. x = 2, y = 5 16. c = 2,d = −3

1. p = −1, q = −2 2. x = 4, y = 6 3. a = 2, b = 3 4. s = 4, t = −1 5. x = 3, y = 4 6. u = 12, v = 2 7. x = 10, y = 15 8. a = 0.30,b = 0.40

1. x = 1 2 , y = 1

4 2. a = 1

3 ,b = −1

3. p = 1 4 ,q = 15 4. x = 10, y = 5

5. c = 3,d = 4 6. r = 3, s = 1 2

7. x = 5, y = 13 4

8. 1

1. a = 0.2,b = 4 2. I1 = 6.47, I2 = 4.62 3. u = 12,a = 4, v = 26 4. £15500,£12800 5. m = −0.5,c = 3 6. α = 0.00426, R0 = 22.56 7. a = 12,b = 0.40 8. a = 4,b = 10 9. F1 = 1.5, F2 = −4.5

1. x = 2, y = 1, z = 3 2. x = 2, y = −2, z = 2 3

5. x x = 1, y = 6, z = 7 7. x x = −4, y = 3, z = 2 9. x = 1.5, y = 2.5, z = 4.5

10. i1 = −5, i2 = −4, i3 = 2 11. F1 = 2, F2 = −3 F3 = 4

1. 4 or −4 2. 4 or −8 3. 2 or −6 4. −1.5 or 1.5 5. 0 or −4

3 6. 2 or −2

7. 4 8. −5 9. 1 10. −2 or −3 11. −3 or −7 12. 2 or −1 13. 4 or −3 14. 2 or 7 15. −4 16. 2 17. −3 18. 3 or −3

19. −2 or −2 3

20. −1.5 21. 1 8

or −1 8

22. 4 or −7 23. −1 or 1.5 24. 1 2

or 1 3

25. 1 2

or −4 5

26. 11 3

or −1 7

27. 3 8

or −2

28. 2 5

or −3 29. 4 3

or −1 2

30. 5 4

or −3 2

31. x2 − 4x + 3 = 0 32. x2 + 3x − 10 = 0 33. x2 + 5x + 4 = 0 34. 4x2 − 8x − 5 = 0 35. x2 − 36 = 0 36. x2 − 1.7x − 1.68 = 0

1. −3.732 or −0.268 2. −3.137 or 0.637 3. 1.468 or −1.135 4. 1.290 or 0.310 5. 2.443 or 0.307 6. −2.851 or 0.351

1. 0.637 or −3.137 2. 0.296 or −0.792 3. 2.781 or 0.719 4. 0.443 or −1.693 5. 3.608 or −1.108 6. 1.434 or 0.232 7. 0.851 or −2.351 8. 2.086 or −0.086 9. 1.481 or −1.081 10. 4.176 or −1.676

11. 4 or 2.167 12. 7.141 or −3.641 13. 4.562 or 0.438

1. 1.191s 2. 0.345A or 0.905A 3. 7.84cm 4. 0.619m or 19.38m 5. 0.0133 6. 1.066m 7. 86.78cm 8. 1.835m or 18.165m 9. 7m

10. 12 ohms,28ohms

1. x = 1, y = 3 and x = −3, y = 7

2. x = 2 5 , y = −1

5 and −12

3 , y = −41

3 3. x = 0, y = 4 and x = 3, y = 1

1. 4 2. 4 3. 3 4. −3 5. 1 3

6. 3 7. 2 8. −2 9. 11 2

10. 1 3

11. 2 12. 10000 13. 100000 14. 9 15. 1 32

16. 0.01 17. 1 16

18. e3

1. log6 2. log15 3. log2 4. log3 5. log12 6. log500 7. log100 8. log6 9. log10 10. log1 = 0 11. log2

12. log243 or log35 or 5 log3 13. log16 or log24 or 4 log2 14. log64 or log26 or 6 log2 15. 0.5 16. 1.5 17. x = 2.5 18. t = 8 19. b = 2 20. x = 2 21. a = 6 22. x = 5

1. 1.690 2. 3.170 3. 0.2696 4. 6.058 5. 2.251 6. 3.959 7. 2.542 8. −0.3272 9. 316.2

1. (a) 0.1653 (b) 0.4584 (c) 22030 2. (a) 5.0988 (b) 0.064037 (c) 40.446 3. (a) 4.55848 (b) 2.40444 (c) 8.05124 4. (a) 48.04106 (b) 4.07482 (c) −0.08286 5. 2.739 6. 120.7m

1. 2.0601 2. (a) 7.389 (b) 0.7408 3. 1− 2x2 − 8

3 x3 − 2x4

4. 2x1/2 + 2x5/2 + x9/2 + 1 3 x13/2

+ 1 12

x17/2 + 1 60

1. 3.95,2.05 2. 1.65,−1.30 3. (a) 28cm3 (b) 116min 4. (a) 70◦C (b) 5 minutes

1. (a) 0.55547 (b) 0.91374 (c) 8.8941 2. (a) 2.2293 (b) −0.33154 (c) 0.13087 3. −0.4904 4. −0.5822 5. 2.197 6. 816.2 7. 0.8274 8. 11.02 9. 1.522 10. 1.485

11. 1.962 12. 3 13. 4 14. 147.9 15. 4.901 16. 3.095 17. t = eb+a lnD = ebea lnD = ebelnDa i.e. t = eb Da

18. 500 19. W = PV ln ( U2 U1

)

1. (a) 150◦C (b) 100.5◦C 2. 99.21kPa 3. (a) 29.32 volts (b) 71.31× 10−6 s 4. (a) 1.993m (b) 2.293m 5. (a) 50◦C (b) 55.45s 6. 30.37N 7. (a) 3.04A (b) 1.46s 8. 2.45mol/cm3 9. (a) 7.07A (b) 0.966s

10. £2424

1. (a) Horizontal axis: 1cm = 4V (or 1cm = 5V), vertical axis: 1cm = 10

(b) Horizontal axis: 1cm = 5m, vertical axis: 1cm = 0.1V

2. (a) −1 (b) −8 (c) −1.5 (d) 5 3. 14.5 4. (a) −1.1 (b) −1.4 5. The 1010 rev/min reading should be 1070 rev/min;

(a) 1000 rev/min (b) 167V

1. Missing values: −0.75,0.25,0.75,2.25,2.75; Gradient = 1

2 2. (a) 4,−2 (b) −1,0 (c) −3,−4 (d) 0,4

3. (a) 2, 1 2

(b) 3,−21 2

, 1 2

4. (a) 6,−3 (b) −2,4 (c) 3,0 (d) 0,7

5. (a) 2,−1 2

(b) −2 3 ,−12

3 (c) 1

18 ,2 (d) 10,−42

6. (a) 3 5

(b) −4 (c) −15 6

7. (a) and (c), (b) and (e) 8. (2, 1) 9. (1.5, 6) 10. (1, 2)

11. (a) 89cm (b) 11N (c) 2.4 (d) l = 2.4W + 48 12. P = 0.15W + 3.5 13. a = −20,b = 412

1. (a) 40◦C (b) 128 2. (a) 850 rev/min (b) 77.5V 3. (a) 0.25 (b) 12 (c) F = 0.25L + 12

(d) 89.5N (e) 592N (f) 212N 4. −0.003,8.73 5. (a) 22.5m/s (b) 6.43s (c) v = 0.7t +15.5 6. m = 26.9L − 0.63 7. (a) 1.31 t (b) 22.89% (c) F = −0.09W + 2.21 8. (a) 96× 109 Pa (b) 0.00022 (c) 28.8× 106 Pa

9. + 6 (d) 12N (e) 65N

10. a = 0.85,b = 12,254.3kPa,275.5kPa,280K

1. (a) y (b) x2 (c) c (d) d 2. (a) y (b) √x (c) b (d) a 3. (a) y (b) 1

x (c) f (d) e 4. (a) y

x (b) x (c) b (d) c

5. (a) y x

(b) 1 x2

1. (a) lg y (b) x (c) lga (d) lgb 2. (a) lg y (b) lgx (c) L (d) lgk 3. (a) ln y (b) x (c) n (d) lnm 4. I = 0.0012V2,6.75 candelas 5. a = 3.0,b = 0.5 6. a = 5.7,b = 2.6,38.53,3.0 7. R0 = 26.0,c = 1.42 8. y = 0.08e0.24x 9. T0 = 35.4N,μ = 0.27,65.0N,1.28 radians

1. x = 2, y = 4 2. x = 1, y = 1 3. x = 3.5, y = 1.5 4. x = −1, y = 2 5. x = 2.3, y = −1.2 6. x = −2, y = −3 7. a = 0.4,b = 1.6

1. (a) Minimum (0, 0) (b) Minimum (0,−1) (c) Maximum (0, 3) (d) Maximum (0,−1)

2. −0.4 or 0.6 3. −3.9 or 6.9 4. −1.1 or 4.1 5. −1.8 or 2.2 6. x = −1.5 or −2, Minimum at (−1.75,−0.1) 7. x = −0.7 or 1.6 8. (a) ±1.63 (b) 1 or −0.3

9. 10. x =

1. x = 4, y = 8 and x = −0.5, y = −5.5 2. (a) x = −1.5 or 3.5 (b) x = −1.24 or 3.24

1. x = −2.0,−0.5 or 1.5 2. x = −2,1 or 3, Minimum at (2.1,−4.1),

Maximum at (−0.8,8.2) 3. x = 1 4. x = −2.0,0.4 or 2.6 5. x = 0.7 or 2.5 6. x = −2.3,1.0 or 1.8 7. x = −1.5

1. 82◦27′ 2. 27◦54′ 3. 51◦11′ 4. 100◦6′52′′ 5. 15◦44′17′′ 6. 86◦49′1′′ 7. 72.55◦ 8. 27.754◦ 9. 37◦57′ 10. 58◦22′52′′

1. reﬂex 2. obtuse 3. acute 4. right angle 5. (a) 21◦ (b) 62◦23′ (c) 48◦56′17′′ 6. (a) 102◦ (b) 165◦ (c) 10◦18′49′′ 7. (a) 60◦ (b) 110◦ (c) 75◦ (d) 143◦ (e) 140◦

(f ) 20◦ (g) 129.3◦ (h) 79◦ (i) 54◦ 8. Transversal (a) 1 & 3, 2 & 4, 5 & 7, 6 & 8

(b) 1 & 2, 2 & 3, 3 & 4, 4 & 1, 5 & 6, 6 & 7, 7 & 8, 8 & 5, 3 & 8, 1 & 6, 4 & 7 or 2 & 5 (c) 1 & 5, 2 & 6, 4 & 8, 3 & 7 (d) 3 & 5 or 2 & 8

9. 59◦20′ 10. a = 69◦,b = 21◦,c = 82◦ 11. 51◦ 12. 1.326 rad 13. 0.605 rad 14. 40◦55′

1. (a) acute-angled scalene triangle (b) isosceles triangle (c) right-angled triangle (d) obtuse-angled scalene triangle (e) equilateral triangle (f) right-angled triangle

2. a ◦, d ◦

3. DF, DE 4. 52◦ 5. 122.5◦

6. φ = 51◦, x = 161◦ 7. 40◦,70◦,70◦,125◦, isosceles 8. a = 18◦50′,b = 71◦10′,c = 68◦,d = 90◦,

e = 22◦, f = 49◦, g = 41◦ 9. a = 103◦,b = 55◦,c = 77◦,d = 125◦,

e = 55◦, f = 22◦, g = 103◦,h = 77◦, i = 103◦, j = 77◦,k = 81◦

10. 17◦ 11. A = 37◦, B = 60◦, E = 83◦

1. (a) congruent BAC, DAC (SAS) (b) congruent FGE, JHI (SSS) (c) not necessarily congruent (d) congruent QRT, SRT (RHS) (e) congruent UVW, XZY (ASA)

2. proof

1. x = 16.54mm, y = 4.18mm 2. 9cm,7.79cm 3. (a) 2.25cm (b) 4cm 4. 3m

1-5. Constructions – see similar constructions in worked problems 30 to 33 on pages 179-180.