Thomas Calculus 13th [Solutions]

Section 3.10 Related Rates 189
27.
2
2
3 2
V 1 r h, h 3 (2 r)
3r r 4h V 1 4h h 16 h dV 16 h dh
3 8 4 3 3 3 27 dt 9 dt
(a) dh
9 (10) 90 0.1119 m/sec 11.19 cm/sec
dt 2
h 4 16 4 256
(b) r 4h
dr 4 dh 4 90 15 0.1492 m/sec 14.92 cm/sec
3 dt 3 dt 3 256 32
2
2
3
28. (a) V 1 r h and r 15h
V 1 15h
h 75 h
3
2 3 2 4
8 0.0113 m/min 1.13 cm/min
225
(b) r 15h
dr 15 dh dr 15 8 4
2 dt 2 dt dt h 5 2 225 15
dV 225
2
h dh dh 4( 50)
dt 4 dt dt h 5
2
225 (5)
0.0849 m/sec 8.49 cm/sec
2
29. (a) V y (3 R y) dV [2 (3 )
3
dt 3 y R y 2 dy dy
( 1)]
2
1
y
dt dt 3 (6 Ry 3 y ) dV at R 13 and y 8
dt
dy
we have 1 ( 6) 1 m/min
dt 144 24
(b) The hemisphere is one the circle 2 (13 ) 2
2
r y 169 r 26 y y m
2 1/2
(c) r (26 y y ) dr 1 2 1/2
dt 2 (26 y y ) (26 2 ) dy dr 13 y dy
y
dt dt
2
26 y y
dt
dr
dt
y 8
13 8 1
26 8 64 24
5
288 m/min
30.
4 3 2
If V r , S 4 , and dV
2
r
kS 4 k r , then dV
3
dt
dt
Therefore, the radius is increasing at a constant rate.
2 2
4 r dr 4k r
dt
2
4 r dr
dt
dr
dt
k , a constant.
31.
4 3
If V r , r 5, and dV
3
100 ft /min, then dV 2
4 r dr dr 1 ft/min. Then S
3
dt
dt dt dt
dS
2
8 r dr 8 (5)(1) 40 ft /min, the rate at which the surface area is increasing.
dt dt
2
4 r
32. Let s represent the length of the rope and x the horizontal distance of the boat from the dock.
(a) We have s
2 x
2 36 dx s ds s ds . Therefore, the boat is approaching the dock at
dt x dt
s
2 36
dt
dx 10 ( 2) 2.5 ft/sec.
dt
2
s 10 10 36
(b) cos 6 sin d 6 dr d 6 dr . Thus, r 10, x 8, and sin 8
r dt 2 dt dt 2
r
r sin dt
10
d 6 ( 2) 3 rad/sec
dt 2 8
10
20
10
33. Let s represent the distance between the bicycle and balloon, h the height of the balloon and x the horizontal
distance between the balloon and the bicycle. The relationship between the variables is s 2 h 2 x
2
ds 1 h dh x dx ds 1 [68(1) 51(17)] 11 ft/sec.
dt s dt dt dt 85
34. (a) Let h be the height of the coffee in the pot. Since the radius of the pot is 3, the volume of the coffee is
(b)
V 9 h dV 9 dh the rate the coffee is rising is dh 1 dV 10 in/min.
dt dt
dt 9 dt 9
2
3
Let h the height of the coffee in the pot. From the figure, the radius of the filter r h V 1 r h h
2 3 12 ,
the volume of the filter. The rate the coffee is falling is dh 4 dV 4 ( 10) 8 in/min.
dt 2
h dt 25 5
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