Question

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 8 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 22 feet high

Answer 1

Answer:

(11π/9 )ft/s

Step by step Explanation

Let us denote the height as h ft

But we were told that The diameter of the base of the cone is approximately three times the altitude, then

Let us denote the diameter = 3h ft, and the radius is 3h/2

The volume of the cone is

V = (1/3)π r^2 h

Then if we substitute the values we have

= (1/3)π (9h^2/4)(h) = (3/4)π h^3

dV/dt = (9/4)π h^2 dh/dt

We were given as 22feet and rate of 8 cubic feet per minute

h = 22

dV/dt = 8

8= (9/4)π (22) dh/dt

= 11π/9ft/s

Therefore, the rate is the height of the pile changing when the pile is 22 feet is

11π/9ft/s