How much work is required to pump all of the water to a spout even with the top of the tank?

Question

A water tank is shaped like a pyramid with its point down. The top base of the pyramid is a square with 8 meter sides. The bottom point of the pyramid is 10 meters below the center of the top base. The tank is filled to a depth of 7 meters. How much work is required to pump all of the water to a spout even with the top of the tank? [use ρ=1000kg/m³ and g=9.8m/s²]

Answer 1

Work = ∫Fdh where F= Weight of a slice of water and h is the distance from that slice to the top. Note that the length of any side will be 8/10h, thus the area of a slice will be .64h² and the volume of a slice will be .64h²dh.Note further that the weight of that slice will be .64ρgh²dh and the work needed to move that slice to the top = 6272h³dh. Integrate that from h=0 to h=10 to find the total work is 1.57E6 Joules