Let h(t) and V(t) be the height and volume of water in a tank at time t. If water drains through a hole with area a at the bottom of the tank, then Torricelli’s Law says that
where g is the acceleration due to gravity. So the rate at which water flows from the tank is proportional to
the square root of the water height.
Part 1
Suppose the tank is cylindrical with height 6 feet and radius 2 feet and the hole is circular with radius 1 inch. If
we take , show that h satisfies the differential equation
You are going to want to start with the formula for volume of a cylinder then use the equation given above for
Torricelli’s Law to substitute in. Your work should start with the volume formula and end with the differential
equation above.