Find the work done in pumping the water over the rim of a tank that is 50 feet long and has a semicircular end of radius 10 feet if the tank is filled to a depth of 7 feet.

Question

Find the work done in pumping the water over the rim of a tank that is 50 feet long and has a semicircular end of radius 10 feet if the tank is filled to a depth of 7 feet.

Answer 1

Elementary work for pumping out section on height x feet from the bottom dW = 2√(100 - x²)·50(10 - x)·62.4·dx;

W = ∫₀⁷6240(10 - x)√(100 - x²)dx = (x = 10sint; dx = 10costdt) = 6240000∫₀^arcsin0.7(1 - sint)cos²tdt =

6240000∫₀^arcsin0.7[(1/2 + cos2t/2)dt + cos²t·d(cost)] = 6240000(t/2 + sin2t/4 + cos³t/3)₀^{arcsin n0.7}=

6240000(0.63765 + 0.38392) = 2656992 lb·feet