Question

Pumping Water A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty (a) half of the tank? (b) all of the tank?

Short answer

The work done to pump half of the tank is 9984 ft-lbs and to pump all of the tank is 19968 ft-lbs.

Step-by-step solution

Calculate volume of the tank

The formula for the volume of a rectangular prism (tank) is length × width × height. Therefore, the volume of the tank is: \(V = 4 \,ft \times 5\, ft \times 4\, ft = 80 \,ft^3\)

(a): Determine work to pump half of the tank

The volume of half of the tank is \(V = \frac{80 \,ft^3}{2} = 40\, ft^3\). The weight of the water is the density times the volume, thus the weight of the water with its density of 62.4 lbs/ft^3 is \( W = 40 \,ft^3 \times 62.4 \,lbs/ft^3 = 2496\, lbs\). The Work done to pump half the tank is 'Weight x Height' as each cubic foot of water is pumped the full 4 feet to the top of the tank. Therefore, the work done is \(Work = 2496\, lbs \times 4\, ft = 9984\, ft-lbs\)

(b): Determine work to pump all of the tank

The weight of all the water in the tank is \( W = 80 \,ft^3 \times 62.4 \,lbs/ft^3 = 4992\, lbs\). Therefore, the work done to drain the whole tank is \(Work = 4992\, lbs \times 4\, ft = 19968\, ft-lbs\)