Question

Sand falls from an overhead bin, accumulating in a conical pile with a radius that is always three times its height. If the sand falls from the bin at a rate of 120 (ft^3)/min, how fast is the height of the sandpile changing when the pile is 10 feet high? [V =1/3(r^2)h]

Answer 1

V = (1/3)Pi*(r^2)h

r = 3h => V = (1/3)Pi*[(3h)^2]h = 3Pi*(h^3)

dV / dt = 9Pi(h^2)*[dh/dt]

dh/dt = [dV/dt] / [9Pi(h^2)]

dV/dt = 120 (ft^3)/min
h= 10 ft

dh/dt = [120(ft^3)/min] / [9Pi(10ft)^2] = 0.042 ft/min

Answer: 0.042 ft/min