At a sand and gravel pit sand is falling off a conveyor and

At a sand and gravel pit, sand is falling off a conveyor and onto a conical pile at a rate of 10 Cubic feet per minute. The diameter of the base of the cone is three times the altitude of the cone. At what rate is the height of the pile changing when the pile is 15 feet high?

Solution

D = 3h

V = (pi/3)r^2h
V = (pi/3)(D/2)^2h
V = (pi/3)(3h/2)^2h
V = (pi/3)(9h^2/4)h
V = (3pi/4)h^3

dV/dt = (3pi/4)(3h^2)(dh/dt)
dV/dt = (9pi/4)(h^2)(dh/dt)
dh/dt = 4(dV/dt)/(9pi * h^2)

h = 15
dV/dt = 10

dh/dt = 4(10)/(9pi * 15^2)
dh/dt = 4(2)/(9pi * 45)
dh/dt = 8/(405pi)