Unformatted text preview: During the time interval
0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate W ( t ) = 95 t sin 2 ( 6t ) gallons per hour. During the same time interval, water is removed from the tank at the rate
R( t ) = 275sin 2 ( 3t ) gallons per hour. (a) Is the amount of water in the tank increasing at time t = 15 ? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time t = 18 ?
(c) At what time t, for 0 ≤ t ≤ 18, is the amount of water in the tank at an absolute minimum? Show the
work that leads to your conclusion.
(d) For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R( t )
until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not
solve, an equation involving an integral expression that can be used to find the value of k.
(a) No; the amount of water is not increasing at t = 15
since W (15 ) − R (15 ) = −121.09 < 0. (b) 1200 + 18 ∫0 (W ( t ) − R( t ) ) dt = 1309.788 ⎧ 1 : limits
⎪
3 : ⎨ 1 : integrand
⎪ 1 : answer
⎩ 1310 gallons (c) W ( t ) − R( t ) = 0
t = 0, 6.4948, 12.9748 ⎧ 1 : interior critical points
⎪ 1 : amount of water is least at
⎪
3: ⎨
t = 6.494 or 6.495
⎪
⎪
⎩ 1 : analysis for absolute minimum t (hours) gallons of water 0
6.495
12.975
18 1 : answer with reason 1200
525
1697
1310 The values at the endpoints and the critical points
show that the absolute minimum occurs when
t = 6.494 or 6.495. (d) k ∫18 R( t ) dt = 1310 ⎧ 1 : limits
2: ⎨
⎩ 1 : equation Copyright © 2005 by College Board. All rights reserved.
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This note was uploaded on 10/01/2009 for the course OC 9876 taught by Professor Dq during the Spring '09 term at UC Merced.
 Spring '09
 Dq

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