576
Chapter 5 | Integration
230. A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36 π in. 3 to 288 π in. 3 between time t =30 and t =60 seconds, find the net change in the radius of the balloon during that time. 231. Water flows into a conical tank with cross-sectional area πx 2 at height x and volume πx 3 3 up to height x . If water flows into the tank at a rate of 1 m 3 /min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min. 232. A horizontal cylindrical tank has cross-sectional area A ( x ) =4 ⎛ ⎝ 6 x − x 2 ⎞ ⎠ m 2 at height x meters above the bottom when x ≤3. a. The volume V between heights a and b is ∫ a b A ( x ) dx . Find the volume at heights between 2 m and 3 m. b. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, dx dt = dx dV dV dt , at how many meters per minute is the height of oil in the tank changing, expressed in terms of x , when the height is at x meters? c. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?
233. The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical 24-hour period, with hour 1 corresponding to midnight to 1 a.m. Hour Power Hour Power
1
28
13
48
2
25
14
49
3
24
15
49
4
23
16
50
5
24
17
50
6
27
18
50
7
29
19
46
8
32
20
43
9
34
21
42
10
39
22
40
11
42
23
37
12
46
24
34
Find the total amount of energy in gigawatt-hours (gW-h) consumed by the city in a typical 24-hour period.
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