At what rate is the height of the pile changing when the pile is 15ft hight?

Question

At a sand and gravel plant, sand is falling off a conveyor onto a conical pile at the rate of 10 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 15ft hight?

Answer 1

Let r = radius of base at time t

     h = height at time t

     V = volume of cone at time t

 

Given:  dV/dt = 10

Find: dh/dt when h = 15

 

V = (1/3)πr²h   Since 2r = 3h, r = (3/2)h

 

V = (1/3)π[(3/2)h]²h

 

   = (3/4)πh³

 

dV/dt = (9/4)πh²(dh/dt)

 

10 = (9/4)π(15)²(dh/dt)

 

dh/dt = 8/(405π) ft/min