please help with this

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 equal. How fast is the height of the pile increasing when the pile is 24 feet high?

Answer 1

The relevant equations are that the height = diameter or h = 2r and the volume = 1/3 π r^2 * h.

Substituting in the r = h/2 into the volume we get V = 1/3 π (h/2)^2 * h or V = 1/3 pi h^3/4

Simplifying we get V = 1/12 π h^3. Taking the derivative with respect to time we get

dV/dt = 1/4 π h^2 dh/dt

We were given dV/dt = 40 ft³/min and h = 24 ft. Plugging these in we get 40 = 1/4 π (24)^2 dh/dt.

Solving for dh/dt we get 160/24² π ft/min or 5/(18π) ft/min which is approximately 0.0884194128288 ft/min

Hope this helps!