a) We are going to pump all the water from the tank and lift it to a specified height. So let's take this height as our reference level for potential energy. We are going to lift all this water up to the reference height, so the work done will simply be equal to the negative of the potential energy stored in the water.

So how to find the potential energy of the water? Imagine the cylinder as a series of disks, each with a height dh. Each disk will have a volume of:

dV = (pi)r^2 * dh

Thus each disk will contain a mass of water equal to:

dm = (rho) * dV (where rho is the density of water.)

dm = (rho)*(pi)r^2 * dh

Thus the potenial energy of the disk will be:

d(PE) = dm * g * h, where h is the height of the disk.

d(PE) = g*(rho)*(pi)*r^2 * hdh

Now integrate over the height coordinate of the cylinder:

Int(d(PE)) = PE = Int(g*(rho)*(pi)*r^2 * hdh, h, 0, -H) where H is the height of the cylinder. (Recall that we require + to be upward, so we are integrating the cylinder from 0 to -H: we are integrating "upside down.")

PE = -(1/2)g*(rho)*(pi)*r^2*H^2

Thus W = -PE = -(1/2)g*(rho)*(pi)*r^2*H^2

b) This is the same idea as a) except that now we are pumping the water up 5 ft more. So add a Mg(5 ft) to the answer in a) where M is the total mass of the water in the cylinder.

-Dan