Question

A water heater in the shape of a circular cylinder is standing vertically, resting on one of the round ends. The tank is 60 inches tall and has a volume of 30 gallons. If water is flowing into the tank at the rate of 5 gallons per minute, at what rate is the level of the water in the tank increasing?

Answer 1

the volume of a 1" layer of water is 1/2 gal. So, when the height of the water is h inches, the volume of water is

v = 1/2 h
dv/dt = 1/2 dh/dt
5 = 1/2 dh/dt
so,
dh/dt = 10 in/min

You can go through all that pi r^2 stuff, but it's not needed here.