Question

Gravel is being dumped from a conveyor belt at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 22 feet high?

Here are the steps and reasoning I took, however arrived at an incorrect answer.

so the derivative of the volume of a cone is:

dv/dt = 1/3π(2rh dr/dt + r^2 dh/dt)

I know r = 11 and since the base diameter doesn't change, dr/dt = 0 so I can ignore the first term in the parentheses.

so dv/dt = 1/3π * r^2 dh/dt

substituting gives:

40 = 1/3π * 121 dh/dt

dh/dt = 0.3156

but the answer is 0.1052

Can someone explain what was wrong with my reasoning?

Answer 1

Never mind, I guess I misinterpreted the problem because the radius actually does change