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AP Calculus Question of the Month: April
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by Ben Klein
Davidson College
Davidson, North Carolina
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Revisiting the Leaky Tank from 2000 AB4
Question AB4 from the 2000 AP Calculus Examination, which is reproduced in slightly modified form below, involved a leaky tank. As in the January 2004 Question of the Month, we are going to propose some additional parts that follow naturally from the parts that actually appeared in the problem. The new parts will give students an opportunity to work with a function that depends on a parameter and will also show how small changes in a mathematical model can produce big changes in the behavior of its solution.
Remember that you can download solutions, scoring guidelines, and sample responses for the original problem right here on AP Central, but we will provide solutions for all the parts since we have modified the original problem slightly.
AB4
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of 
gallons per minute for 
minutes until the tank is first empty. [The original problem had a fixed upper limit for t.] At time t = 0, the tank contains 30 gallons of water.
(A) How many gallons of water leak out of the tank from time t = 0 until t = 3 minutes?
(B) How many gallons of water are in the tank at time t = 3 minutes?
(C) Write an expression for A(t), the total number of gallons of water in the tank at time t.
(D) At what time t is the amount of water in the tank a maximum? Justify your answer. What is the maximum amount of water in the tank? [This second question did not appear in the original problem.]
The following parts, (E) through (I), were not part of the problem as it appeared on the AP Examination.
(E) At what time t is the tank first empty?
In the following two parts, (F) and (G), assume that water is added to the tank at the constant rate of r gallons per minute, where r is a positive parameter, and that nothing else about the problem changes.
(F) Find the value of r such that the tank is first empty at t = 120 minutes, and for this value of r, determine the maximum amount of water in the tank.
(G) Find the value of r such that the amount of water in the tank is a maximum at t = 120 minutes, and for this value of r, determine (1) the maximum amount of water in the tank and (2) the time at which the tank first becomes empty.
In the following two parts, (H) and (I), assume that the rate at which water is added to the tank at time t minutes is rยทt gallons per minute and that nothing else about the problem changes. Thus, the rate at which water is added to the tank increases over time, as does the rate at which water leaves the tank. You might want to speculate about the consequences of this before tackling these next two parts. You may also want to find an expression for A(t), the total number of gallons of water in the tank at time t before working on parts (H) and (I). The expression you find will be very similar to the one used in parts (F) and (G) above.
(H) Let r = 1/4 and show that the number of gallons of water in the tank does not have a maximum and the tank is never empty. Determine the minimum amount of water in the tank.
If you graph A(t) with r = 1/10, for example, you will find that the tank empties out at approximately t = 15 minutes. Thus, with r = 1/10, the minimum amount of water in the tank is zero. This observation sets the stage for the final part of this problem.
(I) Estimate (or even better, find) the largest value of r for which the tank does become empty.
Click here to view the answers!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. He just completed a four-year term on the AP Calculus Development Committee.
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