How fast is the height of the pile increasing when the pile is 12 feet high?

Question

Gravel is being dumped from a conveyor belt at a rate of 50 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 12 feet high?

Answer 1

diameter = 2r = h --> r = h/2 , dV/dt = 50 ft³/min

Volume, V = (1/3)πr²h = (2/3)π(h/2)²h = (2/3)πh³/4 = πh³/6

dV/dt = 3πh²/6 dh/dt = (π/2)h² dh/dt --> dh/dt = 2(dV/dt)/(πh²)

When h = 12:

dh/dt = 2(50)/(144π) = 25/(36π) ft/min