Gravel is being dumped from a conveyor

Question

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 the same. How fast is the height of the pile increasing when the pile is 10 feet high?

 

When the pile is 10 feet high, it's height is increasing at _____ feet per minute.

Answer 1

V = Volume of a right circular cone

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

V = (1/3) pi (h/2)^2 h Diameter = height, so r = h/2

V = (1/3) pi (1/4) h^3

V = (1/12) pi h^3

d/dt [V] = d/dt [ (1/12) pi h^3 ]. Take derivative of both sides.

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

V' = (pi/4) h^2 h'

30 = (pi/4)*(10^2)*h' Use given information

30 = (pi/4)*100*h'

30 = 25*pi*h'

30/(25*pi) = h'

dh/dt = 6/(5*pi)

dh/dt ~ 0.382 feet/min When pile is 10 feet high, it's increasing at this rate.