Calc 1 Related Rates

Question

Gravel is being dumped from a conveyor belt at a rate of 50 feet cubed 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 14 ft high?

Answer 1

Let h represent the height of the pile at time t. The volume, V, of the pile at time t is then V = 1/3 π (h/2)² * h = (π/12)h³. It is given dV/dt = 50, so (π/12)(3h²) dh/dt = 50, or dh/dt = 200π/h². When h = 14 ft the pile is increasing at a rate of dh/dt = 200π/(14²) ≈ 3.2 ft/min.