[Related Rates] Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute.

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 17 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by ²   

Answer 1

Since h = r (always according the the given), the formula for volume can be written:

V = 1/3 πh³ (we want h in the formula since we want to solve for dh/dt). We are given that dV/dt = 50 ft³/min.

Differentiating with respect to t:

dV/dt = πh²dh/dt (using chain rule when finding derivative of h with respect to t)

Substituting:

50 ft³/min = π(17 ft)²dh/dt

Solving for dh/dt:

dh/dt = 50 ft³/min / 289 π ft = 50/289π ft/min