Question

A fuel oil tank is an upright cylinder, buried so that its circular top is 14 feet beneath ground level. the tank has a radius of 5 feet and is 15 feet high, although the current oil level is only 14 feet deep. calculate the work required to pump all of the oil to the surface. oil weighs 50lb/ft350lb/ft3.

Answer 1

Say we have a cylinder that has a height of dx, we see that the cylinder has a volume of: 

Vcylinder = πr^2*h = π(5)^2(dx) = 25π dx

Then, the weight of oil in this cylinder is: 

Fcylinder = 50 * Vcylinder = (50)(25π dx) = 1250π dx. 

Then, since the oil x feet from the top of the tank needs to travel x feet to get the top, we have: 

Wcylinder = Force x Distance = (1250π dx)(x) = 1250π x dx. 

Integrating from x1 to x2 ft gives the total work to be: (x1 = distance from top liquid level to ground level; x2 = distance from bottom liquid level to ground level)

W = ∫ 1250π x dx  
W = 1250π ∫ x dx
W = 625π * (x2 – x1)

x2 = 14 ft  + 15 ft = 29 ft

x1 = 14 ft + 1 ft = 15 ft

W = 625π * (29^2 - 15^2) 
W = 385,000π ft-lbs = 1,209,513.17 ft-lbs