How fast is the height of the pile increasing when the pile is 22 feet high?

Question

Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 22 feet high?

Recall that the volume of a right circular cone with height h and radius of the base r is given by V=13πr2h

Answer 1

Well diameter = height so, 2r=h, so r = h/2, so V = (1/12) πh³

dV/dt = 3 (1/12) π h² dh/dt

dV/dt = 30, h = 22

so 30 = 3 (1/12) π 22² dh/dt

dh/dt = 120 / (22²π)