348 Basic Engineering Mathematics
9. (a)
1
5
(b) 6 (c) E =
1
5
L + 6 (d) 12 N (e) 65 N
10. a = 0.85, b = 12, 254.3 kPa, 275.5 kPa, 280 K
Chapter 18
Exercise 70 (page 149)
1. (a) y (b) x
2 (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
x 2 (c) a (d) b
6. a = 1.5, b = 0.4, 11.78 mm 2 7. y = 2x 2 + 7, 5.15
8. (a) 950 (b) 317 kN
9. a = 0.4, b = 8.6 (i) 94.4 (ii) 11.2
Exercise 71 (page 154)
1. (a) lg y (b) x (c) lg a (d) lg b
2. (a) lg y (b) lg x (c) L (d) lg k
3. (a) ln y (b) x (c) n (d) ln m
4. I = 0.0012 V
2
, 6.75 candelas
5. a = 3.0, b = 0.5
6. a = 5.7, b = 2.6, 38.53, 3.0
7. R 0 = 26.0, c = 1.42
8. y = 0.08e 0.24x
9. T 0 = 35.4 N, μ = 0.27, 65.0 N, 1.28 radians
Chapter 19
Exercise 72 (page 156)
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
Exercise 73 (page 160)
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. (−2.6, 13.2), (0.6, 0.8); x = −2.6 or 0.6
10. x = −1.2 or 2.5 (a) −30 (b) 2.75 and −1.50
(c) 2.3 or −0.8
Exercise 74 (page 161)
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
(c) x = −1.5 or 3.0
Exercise 75 (page 162)
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
Chapter 20
Exercise 76 (page 167)
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
Exercise 77 (page 169)
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
Exercise 78 (page 173)
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
9. (a)
1
5
(b) 6 (c) E =
1
5
L + 6 (d) 12 N (e) 65 N
10. a = 0.85, b = 12, 254.3 kPa, 275.5 kPa, 280 K
Chapter 18
Exercise 70 (page 149)
1. (a) y (b) x
2 (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
x 2 (c) a (d) b
6. a = 1.5, b = 0.4, 11.78 mm 2 7. y = 2x 2 + 7, 5.15
8. (a) 950 (b) 317 kN
9. a = 0.4, b = 8.6 (i) 94.4 (ii) 11.2
Exercise 71 (page 154)
1. (a) lg y (b) x (c) lg a (d) lg b
2. (a) lg y (b) lg x (c) L (d) lg k
3. (a) ln y (b) x (c) n (d) ln m
4. I = 0.0012 V
2
, 6.75 candelas
5. a = 3.0, b = 0.5
6. a = 5.7, b = 2.6, 38.53, 3.0
7. R 0 = 26.0, c = 1.42
8. y = 0.08e 0.24x
9. T 0 = 35.4 N, μ = 0.27, 65.0 N, 1.28 radians
Chapter 19
Exercise 72 (page 156)
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
Exercise 73 (page 160)
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. (−2.6, 13.2), (0.6, 0.8); x = −2.6 or 0.6
10. x = −1.2 or 2.5 (a) −30 (b) 2.75 and −1.50
(c) 2.3 or −0.8
Exercise 74 (page 161)
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
(c) x = −1.5 or 3.0
Exercise 75 (page 162)
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
Chapter 20
Exercise 76 (page 167)
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
Exercise 77 (page 169)
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
Exercise 78 (page 173)
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