calculas homework help.

Question

The h in the formula for velocity is lower then the 2

Gravel is being dumped from a conveyor belt at a rate of 30 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 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=1/3πr^2h

Answer 1

We are told that the "base diameter and height are always equal" meaning that d = h. Since d/2 = r, then:

r = h/2

So the equation V = (1/3)πr²h becomes V = (1/3)π(h/2)²(h) or:

V(h) = (π/12)h³

We are also told that the volume is increasing at "a rate of 30 cubic feet per minute". That is dV/dt.

Taking the derivative of V(h) with respect to time (t), using the chain rule, we get:

dV/dt = (3π/12)h²•dh/dt

dV/dt = (π/4)h²(dh/dt)

Solving for dh/dt (since we are trying to determine "how fast is the height of the pile increasing") we get:

dh/dt = 4(dV/dt)/(πh²) then plugging in 30 for dV/dt and 22 for h we get:

dh/dt = 4(30)/(π22²) = 0.0789 ft/min