At what rate is the height of the pile changing when the pile is 15 feet high?

Question

Other given information:

At a sand and gravel plant, sand is falling off a conveyor belt and onto a conical (cone) pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude.

Answer 1

The volume of a cone is:

V = (1/3)πr²h

1. r = the radius of the base
2. h = height

Since the diameter is 3 times the height, r = 3h/2:

V = (1/3)π(3h/2)²h = 3πh³/4

Take the derivative of V wrt time to find the rate of change:

dV/dt = dV/dh · dh/dt = (9πh²/4)·dh/dt

Where dh/dt = the rate that the height of the pile is changing. We are given dV/dt = 10 and h = 15. Plug those into the dV/dt equation and solve for dh/dt.