Project: A Cylindrical Tank Problem Consider an open cylindrical tank of height h0 mete...

Project: A Cylindrical Tank Problem

Consider an open cylindrical tank of height h0 meters and radius r meters that is filled with water. A circular hole of radius l meters in the bottom of the tank allows the water to flow out under the influence of gravity. According to Torricelli’s law, the water flows out with the same speed that it would acquire in falling freely from the water level in the tank to the hole.

Use Torricelli’s law to derive the following equation for the rate of change of volume of water in the tank,

where h(t) denotes the height of water in the tank at time t, a denotes the area of the hole, and g denotes the acceleration due to gravity. [Hint: First show that an object that is released from rest at a height h hits the ground with a speed .]