Question

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 20 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 12 feet high? (Hint: The formula for the volume of a cone is V = 1/3πr²h.)

Answer 1

let assume that the radius of conical pile is r, diameter is d and height is h,

we have

the volume of the conical pile,

put r = 3h/2,

put dV/dt = 20, h = 12,

he height of the pile increases at rate of 0.01964 feet/min.