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AP Calculus Question of the Month: March 2005
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by Ben Klein
Davidson College
Davidson, North Carolina
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The Coffeepot from 2003 AB5/BC5
This Featured Question is the second in a series devoted to free-response questions from the 2003 AP Calculus Examinations, which were released in early 2005. This Featured Question builds on the "coffeepot problem," which was a common problem on the 2003 AB and BC examinations. The original question combined related rates and separable differential equations. We begin by extending the original question, and then we introduce a similar, but more complicated, situation in which the original cylinder is replaced by a right circular cone.
In the interest of completeness, we will reproduce the "official" solutions for the three parts of the original question along with the solutions for the four new parts.
AB5/BC5
A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at a rate of 
cubic inches per second. (The volume V of a cylinder with radius r and height h is 
.)
(A) Show that 
.
(B) Given that h = 17 at time t = 0, solve the differential equation 
for h as a function of t.
(C) At what time t is the coffeepot empty?
The original question consisted of the three parts (A), (B), and (C) above. The following four parts are new. Part (D) continues our investigation of the cylindrical coffeepot, and the last three parts consider the more complicated conical coffeepot.
(D) Find h as a function of t if the volume V of coffee in the pot is changing at a rate of -5πh cubic inches per second. Assume, as in part (B), that h = 17 when t = 0. Is the coffeepot ever empty under this new assumption?
We now replace the cylindrical coffeepot of the original question with a conical coffeepot whose vertex points downward. The height of the coffeepot is 17 inches, and the radius of its circular top is 5 inches. A sketch of a cross section of the cone is given below. As in the original figure, h represents the depth of the coffee in the pot.
(E) Find h as a function of t if the volume V of coffee in the pot is changing at a rate of 
cubic inches per second. Assume, as in part (B), that h = 17 when t = 0. (That is, the coffeepot is full at time t = 0.) Then decide at what time, if ever, the coffeepot is empty.
(F) Redo part (E) under the assumption that the volume V of coffee in the pot is changing at a rate of 
cubic inches per second.
(G) Suppose that the volume V of coffee in the pot is changing at a rate of 
cubic inches per second, where p is a real number. If, as in part (D), the depth of the coffee in the pot is subject to exponential decay, what is the value of p?
Complete the question before viewing the answers and explanation!
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. In 2003, he completed a four-year term on the AP Calculus Development Committee.
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