Gravel is being dumped from a conveyor belt at a rate of 50 . It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.

Question

Gravel is being dumped from a conveyor belt at a rate of 50 . 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 ft high?

Answer 1

The volume of the conical pile is increasing. Therefore, dV/dt = 50

d = h

How fast is the height of the pile increasing: dh/dt = ?

We want to get the volume formula in terms of h then:

V = 1/3 * pi * r^2 * h

Substitute: r = 1/2 d = 1/2 h

V = 1/3 * pi * (1/2h)^2 * h

V= 1/12 * pi * h^3

Differentiate w.r.t t:

dV/dt = (3)1/12 * pi * h^2 * dh/dt

50 = 1/4 * pi * 24^2 * dh/dt (Solve for dh/dt)

50/ (1/4 * pi * 24^2) = dh/dt

0.1105 ft/s ≈ dh/dt