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

Answer 1

let the height be h ft
then diameter = 3h ft, and the radius is 3h/2

V = (1/3)π r^2 h
= (1/3)π (9h^2/4)(h) = (3/4)π h^3

dV/dt = (9/4)π h^2 dh/dt

when h = 15, dV/dt = 10
10 = (9/4)π (15) dh/dt
dh/dt = 40/(135π) ft/s
= 8π/27 ft/s