EXAMPLE 2.7.3

Gomer has a super-jumbo-sized drip coffee maker. The beverage is produced as hot water filters through a cone-shaped vessel containing coffee grounds (see diagram below)

Assuming that the cone is filled with water, and the water is dripping out at a rate of 10 cu. in. per minute, how long will it take for all of the water to pass through?

SOLUTION

First we need to find the amount of water, in cubic inches, that is contained in the cone. Then we will divide by the rate of 10 cubic inches per minute in order to find the number of minutes required to empty the cone.

Recall the formula for the volume of a right circular cone:

 Volume = (one third) times pi time (radius squared) times height

In this case the diameter of the cone is 1 foot, so the radius is one-half foot; however, since we want to compute volume in cubic inches, not cubic feet, we will convert this measurement to inches:

r = one-half foot = 6 inches

The height of the cone is 3 inches.

Using the formula for volume we have:

Volume = (one third) times pi time (6 squared) times 3 = 36 times pi = 113.1 cubic inches

Now we divide by the rate of 10 cubic inches per minute:

Number of minutes = (113.1 cubic inches) divided by (10 cubic inches per minute) = 11.31 minutes