The work required to pump the water out of the spout.
The work is calculated by using the formula W=∫abf(x)dx
Consider the horizontal slice (shaped like a disk) of water ∆x ft thick and lying at coordinate x.Calculate the radius of the slice by using a similar triangle method
In triangle ∆PQR & ∆TRM
Let x be the depth below the spout
The radius of the disk is r=3816-x ft The volume the disk of water is πr2∆x=π·96416-x2∆x ft3. It weighs about 62.59π6416-x2∆x lb and must be lifted x ft by the pump
Therefore, the work needed to pump the disk water out of the spout is 62.5x9π6416-x2∆x ft-lb The total work required is
The work required to pump the water out of the spout is ≈1.04×105 ft-lb