Question

Raccoon City monitors the height of the water in its cylindrical water tank with an automatic recording device. The tank has a radius of 20 feet. Water is constantly pumped into the tank at a rate of 2400 cubic feet per hour. If the water level was 16.05 feet at 6:59 am and 15.95 feet at 7:01 am, estimate the rate at which the residents of Raccoon City were using water at 7:00 am. Round to the nearest cubic foot per hour.

Answer 1

We have that water is pumped in at 2400 cubic feet per hour.
This means to make order for the water level to remain stable, the water has to be used at the same rate.

We also have that the tank is a cylinder with radius 20 ft.
So one foot of height in the cylinder holds
1(pi)r^2
= pi(20^2)
=400pi cubic feet of water.

Now, to lower the level at a rate of three feet per hour (since dh/dt = -(15.95-16.05)/(2)=-0.05 we know this is what was happening at 7am), you would have to drain out the initial 2400 cubic feet plus 0.05*(400pi) cubic feet of water, for that extra three feet.

2400 + 0.05*(400pi) = 2400 + 20pi
So you would have to drain at a rate of about 2463 cubic feet per hour to have the water level dropping by three feet per hour.