tank and rate of change

Question

(1 point)

A right conical tank with the point oriented down, a height of 13 feet, and a radius of 6 feet has sprung a leak. How fast is the volume of water in the tank changing when the water is 3 feet high and the water level changes at a rate of 4 inches per minute?

Answer 1

V = 1/3 h π r² , where V is the volume , h is the height and r is the radius.

We have to find relation between the radius and the height in order to make the equation of the volume depends on one variable.

The triangle shown in the figure represent half of the cone with radius 6 and height 13

Assume that at height h from the bottom the radius is r

6 /r =13 /h

13r =6 h

r = 6/13 h

V = ( 1/3) h π (6/13)² ( h)²

V = (1/3) ( 6/13)² π h³

dV/dt = (1/3) ( 6/13)² π * 3* h² dh/dt

When dh/dt =4/12 = 1/3 feet/min and h =3 feet

dV/dt = ( 6/13)² π (3)² * 1/3

= 108/169 π 6

13