A gas station stores its gasoline in a tank under the ground. The tank is a cylinder lying horizontally on its side. (In other words, the tank is not standing vertically on one of its flat ends.) If the radius of the cylinder is 4 feet, its length is 12 feet, and its top is 10 feet under the ground, find the total amount of work needed to pump the gasoline out of the tank. (Gasoline weighs 42 lb/ft3.)

Work done

Explanation:

To evaluate the force on each strip first we have to find the mass on each strip

Using x from the top of water Volume of strip is

ΔV=π r² × height

ΔV=π(0.5)²×Δx

Now mass of strip is given by:

Mass=Density × volume

ΔM=(1+kh)×π(0.5)²×Δx

Hence h is measured from surface of water so h=x-0.3 So after substitution the value of mass is:

ΔM=(1+k(x-0.3)×π(0.5)²×Δx)

Now the force due to gravity

ΔF=(1+k(x-0.3)×π(0.5)²×Δx)×g

The work done by water in moving distance x

ΔW=ΔF × x

ΔW=[(1+k(x-0.3)×π(0.5)²×Δx)×g]× x

W= [[(1+k(x-0.3)×π(0.5)²×Δx)×g]× x]dx

Hence the work done=