Lighthouse Initiative for Texas Classrooms

AP* Calculus AB Problem 5, 1995

This problem was one of the six free-response questions on the 1995 AP* Calculus AB exam. Students had a great deal of difficulty in recognizing the similarity of the water in the cone to the cone figure itself. In Part A, they seemed to have difficulty with the wording of the problem. Had it said, "Solve for V in terms of h," a greater number of students probably would have solved the equation correctly. Students also had difficulty in working with the volume formulas and with the simple symbolic manipulation that was required. All of these difficulties occurred well before the students had a chance to reach the calculus stage of the problem. The concepts of proportionality, similarity, volume, and symbolic manipulation are introduced as early as middle school, but perhaps could be approached a little differently in a Pre-AP* situation. In addition to examining the similarity of plane figures, for example, Pre-AP students could also examine the similarity of three-dimensional figures and embedded figures. Instead of just using formulas to calculate area or volume, there could be more emphasis on manipulating the formulas and expressing them in different forms. The same TEKS would be addressed but in a way that will better prepare students for the AP experience.

AP Calculus, Problem 5, Figure 1

5. As shown in the figure above, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base with area 400π square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of (h-12) feet per minute. (The volume V of a cone with radius r and height h is V = 1/3πr2h.)

(a) Write an expression for the volume of water in the conical tank as a function of h.

(b) At what rate is the volume of water in the conical tank changing when h = 3? Indicate units of measure.

(c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h = 3? Indicate units of measure.


Source: Copyright © 2006. The College Board. Reproduced with permission.