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 the same. How fast is the height of the pile increasing when the pile is 24 feet high?

Question

Recall that the volume of a right circular cone with height h and radius of the base r is given by
v=1/3pir^2h

Answer 1

Let the radius be r, so the diameter is 2r
then the height is 2r

V = (1/3)π r^2 h
= (1/3) π r^2 (2r)
= (2/3)π r^3

dV/dt = 2π r^2 dr/dt

when h = 24
2r = 24
r = 12 and dV/dt = 30

30 = 2π (144) dr/dt
dr/dt = 30/(288π) = 5/(48π) ft/min
or appr .033 ft/min

check my arithmetic.