Calculus Question

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 20 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V=(1/3)πr2ℎ.

Answer 1

v = 1/3πr²h = 1/12πh³; dv/dt = 1/4πh²dh/dt; dh/dt = 4dv/dt/(πh²) = 4·40ft³/min/(π·400ft²) = 0.1273 ft/min