Suppose that a storage device uses electricity to pump water into a storage vessel which is shaped like a parabolic trough, with cross-section profile given by the curve z = (1/100)x ⁴ with z = 0 to 81 meters, and with length is L = 200 meters, as shown:

Compute the total amount of energy that can be stored in this storage vessel, using 1 kg/m ³ as the mass density of water. Express answer in Joules. (Hint: The potential energy of a mass m increases by mgh when it is lifted a height h, where g = 9.81 m/s ² . Assume that all of the water was initially at height z = 0 before being pumped into the vessel).

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