Question

Problem 4. A cylindrical tank laying on its side is buried under the ground so that the top of it is 15 feet below ground. The tank is 12 feet long and has a radius of 5 feet. The tank is filled with gasoline which has a density of 42 pounds per cubic foot. Write a Riemann sum to approximate the work required to empty the tank. Then pass to a limit and find the work that would be required.

Answer 1

Heren the tank is laying on the ground
The radius of the tank = 5 ft
It is at a depth of 15 ft
The equation of the circle with the center at x = 0, y = -20 is
x^2+(y+20)^2 = 5^2
Consider a strip of volime with length L, width 2x and thickness dy
The volume of such a strip will be
dV = L*2xdy = L*2*sqrt(25-(y+20)^2)dy
So, the mass will be

Mass = density*Volume

$dM = \rho*L*2*\sqrt{25-(y+20)^2}dy$

The work done to remove this mass is

$dW = dMgh = \rho*L*g*2*\sqrt{25-(y+20)^2}*y dy$

SO, the total work will be

$W = \sum dW =\sum_{y = -25}^{-15}\rho*L*g*2*\sqrt{25-(y+20)^2}*y dy$

This sum can be approximated as an integral.

Giving the values,

$W =\int_{y = -25}^{-15}2*32.2*42*12*\sqrt{25-(y+20)^2}*y dy$

The total work done will be

W = 2.549 *10^7 ft.lb