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

Question

 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 22 feet high?

Recall that the volume of a right circular cone with height h and radius of the base r is given by

V= 1/3π r²h

Answer 1

So you know that the base diameter and height are always equal. That means that d = h, or 2r = h where r is the base radius. Therefore, you can cast the formula for volume in the one relevant variable (h).

V = 1/3*π*(h/2)²*h = (π/12)*h³

Now time differentiate,

dV/dt = (π/4)*h²*(dh/dt)

So dh/dt = 4*(dV/dt)/(π*h²)

I'll leave it to you to plug in the given values.