Activity 6.4.3.
In each of the following situations, determine the total work required to accomplish the described task. In parts (b) and (c), a key step is to find a formula for a function that describes the curve that forms the side boundary of the tank.
(a)
Consider a vertical cylindrical tank of radius 2 meters and depth 6 meters. Suppose the tank is filled with 4 meters of water of mass density 1000 kg/m\(^3\text{,}\) and the top 1 meter of water is pumped over the top of the tank.
(b)
Consider a hemispherical tank with a radius of 10 feet. Suppose that the tank is full to a depth of 7 feet with water of weight density 62.4 pounds/ft\(^3\text{,}\) and the top 5 feet of water are pumped out of the tank to a tanker truck whose height is 5 feet above the top of the tank.
(c)
Consider a trough with triangular ends, as pictured in the following figure where the tank is 10 feet long, the top is 5 feet wide, and the tank is 4 feet deep. Say that the trough is full to within 1 foot of the top with water of weight density 62.4 pounds/ft\(^3\text{,}\) and a pump is used to empty the tank until the water remaining in the tank is 1 foot deep.
