350 Basic Engineering Mathematics
4. t = 122
◦ 7
and 237
◦ 53
5. α = 218 ◦ 41 and 321 ◦ 19
6. θ = 39 ◦ 44 and 219 ◦ 44
Exercise 88 (page 202)
1. 5
2. 180 ◦
3. 30
4. 120 ◦
5. 1, 120 ◦
6. 2, 144 ◦ 7. 3, 90 ◦
8. 5, 720 ◦
9.
7
2
, 960 ◦ 10. 6, 360 ◦ 11. 4, 180 ◦ 12. 5 ms
13. 40 Hz
14. 100 μs or 0.1 ms
15. 625 Hz 16. leading 17. leading
Exercise 89 (page 203)
1. (a) 40 (b) 25 Hz (c) 0.04s or 40 ms (c) 25Hz
(d) 0.29 rad (or 16.62 ◦ ) leading 40 sin 50πt
2. (a) 75 cm (b) 6.37 Hz (c) 0.157 s
(d) 0.54 rad (or 30.94 ◦ ) lagging 75 sin 40t
3. (a) 300 V (b) 100 Hz (c) 0.01 s or 10 ms
(d) 0.412 rad (or 23.61 ◦ ) lagging 300 sin 200πt
4. (a) v = 120 sin100πt volts
(b) v = 120 sin (100πt + 0.43) volts
5. i = 20 sin
80πt −
π
6
A or
i = 20 sin(80πt − 0.524)A
6. 3.2 sin(100πt + 0.488) m
7. (a) 5A, 50 Hz, 20 ms, 24.75
◦ lagging
(b) −2.093A (c) 4.363 A (d) 6.375 ms (e) 3.423 ms
Chapter 23
Exercise 90 (page 207)
1. C = 83 ◦ , a = 14.1 mm, c = 28.9 mm,
area = 189 mm 2
2. A = 52 ◦ 2 , c = 7.568 cm, a = 7.152 cm,
area = 25.65 cm 2
3. D = 19 ◦ 48 , E = 134 ◦ 12 , e = 36.0 cm,
area = 134 cm 2
4. E = 49 ◦ 0 , F = 26 ◦ 38 , f = 15.08 mm,
area = 185.6 mm 2
5. J = 44 ◦ 29 , L = 99 ◦ 31 , l = 5.420 cm,
area = 6.132 cm 2 , or, J = 135 ◦ 31 , L = 8 ◦ 29 ,
l = 0.811 cm, area = 0.917 cm 2
6. K = 47 ◦ 8 , J = 97 ◦ 52 , j = 62.2 mm,
area = 820.2 mm 2 or K = 132 ◦ 52 , J = 12 ◦ 8 ,
j = 13.19 mm, area = 174.0 mm 2
Exercise 91 (page 209)
1. p = 13.2 cm, Q = 47.35 ◦ , R = 78.65 ◦ ,
area = 77.7 cm 2
2. p = 6.127 m, Q = 30.83
◦
, R = 44.17
◦ ,
area = 6.938 m 2
3. X = 83.33 ◦ , Y = 52.62 ◦ , Z = 44.05 ◦ ,
area = 27.8 cm 2
4. Z = 29.77 ◦ , Y = 53.50 ◦ , Z = 96.73 ◦ ,
area = 355 mm 2
Exercise 92 (page 210)
1. 193 km 2. (a) 122.6 m (b) 94.80 ◦ , 40.66 ◦ , 44.54 ◦
3. (a) 11.4 m (b) 17.55 ◦
4. 163.4 m
5. BF = 3.9 m, EB = 4.0 m 6. 6.35 m, 5.37 m
7. 32.48 A, 14.31 ◦
Exercise 93 (page 212)
1. 80.42 ◦ , 59.38 ◦ , 40.20 ◦ 2. (a) 15.23 m (b) 38.07 ◦
3. 40.25 cm, 126.05 ◦
4. 19.8 cm 5. 36.2 m
6. x = 69.3 mm, y = 142 mm 7. 130 ◦ 8. 13.66 mm
Chapter 24
Exercise 94 (page 215)
1. (5.83, 59.04 ◦ ) or (5.83, 1.03 rad)
2. (6.61, 20.82 ◦ ) or (6.61, 0.36 rad)
3. (4.47, 116.57 ◦ ) or (4.47, 2.03 rad)
4. (6.55, 145.58 ◦ ) or (6.55, 2.54 rad)
5. (7.62, 203.20
◦
) or (7.62, 3.55 rad)
6. (4.33, 236.31 ◦ ) or (4.33, 4.12 rad)
7. (5.83, 329.04 ◦ ) or (5.83, 5.74 rad)
8. (15.68, 307.75 ◦ ) or (15.68, 5.37 rad)
Exercise 95 (page 217)
1. (1.294, 4.830)
2. (1.917, 3.960)
3. (−5.362, 4.500)
4. (−2.884, 2.154)
5. (−9.353, −5.400)
6. (−2.615, −3.207)
7. (0.750, −1.299)
8. (4.252, −4.233)
9. (a) 40∠18 ◦ , 40∠90 ◦ , 40∠162 ◦ , 40∠234 ◦ , 40∠306 ◦
(b) (38.04, 12.36), (0, 40), (−38.04, 12.36),
(−23.51, −32.36), (23.51, −32.36)
10. 47.0 mm
4. t = 122
◦ 7
and 237
◦ 53
5. α = 218 ◦ 41 and 321 ◦ 19
6. θ = 39 ◦ 44 and 219 ◦ 44
Exercise 88 (page 202)
1. 5
2. 180 ◦
3. 30
4. 120 ◦
5. 1, 120 ◦
6. 2, 144 ◦ 7. 3, 90 ◦
8. 5, 720 ◦
9.
7
2
, 960 ◦ 10. 6, 360 ◦ 11. 4, 180 ◦ 12. 5 ms
13. 40 Hz
14. 100 μs or 0.1 ms
15. 625 Hz 16. leading 17. leading
Exercise 89 (page 203)
1. (a) 40 (b) 25 Hz (c) 0.04s or 40 ms (c) 25Hz
(d) 0.29 rad (or 16.62 ◦ ) leading 40 sin 50πt
2. (a) 75 cm (b) 6.37 Hz (c) 0.157 s
(d) 0.54 rad (or 30.94 ◦ ) lagging 75 sin 40t
3. (a) 300 V (b) 100 Hz (c) 0.01 s or 10 ms
(d) 0.412 rad (or 23.61 ◦ ) lagging 300 sin 200πt
4. (a) v = 120 sin100πt volts
(b) v = 120 sin (100πt + 0.43) volts
5. i = 20 sin
80πt −
π
6
A or
i = 20 sin(80πt − 0.524)A
6. 3.2 sin(100πt + 0.488) m
7. (a) 5A, 50 Hz, 20 ms, 24.75
◦ lagging
(b) −2.093A (c) 4.363 A (d) 6.375 ms (e) 3.423 ms
Chapter 23
Exercise 90 (page 207)
1. C = 83 ◦ , a = 14.1 mm, c = 28.9 mm,
area = 189 mm 2
2. A = 52 ◦ 2 , c = 7.568 cm, a = 7.152 cm,
area = 25.65 cm 2
3. D = 19 ◦ 48 , E = 134 ◦ 12 , e = 36.0 cm,
area = 134 cm 2
4. E = 49 ◦ 0 , F = 26 ◦ 38 , f = 15.08 mm,
area = 185.6 mm 2
5. J = 44 ◦ 29 , L = 99 ◦ 31 , l = 5.420 cm,
area = 6.132 cm 2 , or, J = 135 ◦ 31 , L = 8 ◦ 29 ,
l = 0.811 cm, area = 0.917 cm 2
6. K = 47 ◦ 8 , J = 97 ◦ 52 , j = 62.2 mm,
area = 820.2 mm 2 or K = 132 ◦ 52 , J = 12 ◦ 8 ,
j = 13.19 mm, area = 174.0 mm 2
Exercise 91 (page 209)
1. p = 13.2 cm, Q = 47.35 ◦ , R = 78.65 ◦ ,
area = 77.7 cm 2
2. p = 6.127 m, Q = 30.83
◦
, R = 44.17
◦ ,
area = 6.938 m 2
3. X = 83.33 ◦ , Y = 52.62 ◦ , Z = 44.05 ◦ ,
area = 27.8 cm 2
4. Z = 29.77 ◦ , Y = 53.50 ◦ , Z = 96.73 ◦ ,
area = 355 mm 2
Exercise 92 (page 210)
1. 193 km 2. (a) 122.6 m (b) 94.80 ◦ , 40.66 ◦ , 44.54 ◦
3. (a) 11.4 m (b) 17.55 ◦
4. 163.4 m
5. BF = 3.9 m, EB = 4.0 m 6. 6.35 m, 5.37 m
7. 32.48 A, 14.31 ◦
Exercise 93 (page 212)
1. 80.42 ◦ , 59.38 ◦ , 40.20 ◦ 2. (a) 15.23 m (b) 38.07 ◦
3. 40.25 cm, 126.05 ◦
4. 19.8 cm 5. 36.2 m
6. x = 69.3 mm, y = 142 mm 7. 130 ◦ 8. 13.66 mm
Chapter 24
Exercise 94 (page 215)
1. (5.83, 59.04 ◦ ) or (5.83, 1.03 rad)
2. (6.61, 20.82 ◦ ) or (6.61, 0.36 rad)
3. (4.47, 116.57 ◦ ) or (4.47, 2.03 rad)
4. (6.55, 145.58 ◦ ) or (6.55, 2.54 rad)
5. (7.62, 203.20
◦
) or (7.62, 3.55 rad)
6. (4.33, 236.31 ◦ ) or (4.33, 4.12 rad)
7. (5.83, 329.04 ◦ ) or (5.83, 5.74 rad)
8. (15.68, 307.75 ◦ ) or (15.68, 5.37 rad)
Exercise 95 (page 217)
1. (1.294, 4.830)
2. (1.917, 3.960)
3. (−5.362, 4.500)
4. (−2.884, 2.154)
5. (−9.353, −5.400)
6. (−2.615, −3.207)
7. (0.750, −1.299)
8. (4.252, −4.233)
9. (a) 40∠18 ◦ , 40∠90 ◦ , 40∠162 ◦ , 40∠234 ◦ , 40∠306 ◦
(b) (38.04, 12.36), (0, 40), (−38.04, 12.36),
(−23.51, −32.36), (23.51, −32.36)
10. 47.0 mm