gravitational potential energy?

Question

A hydroelectric dam holds back a lake of surface area 2.9×106 m^2 that has vertical sides below the water level. The water level in the lake is a height 150 m above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted into electrical energy with 90 % efficiency.

What volume of water must pass through the dam to produce an amount of electrical energy totalling 1000 kilowatt-hours of electrical energy?
What distance does the level of water in the lake fall when this much water passes through the dam?

Answer 1

First note that 1000kWh is a meausre of energy - Joules
E = 1000 x 1000 x 3600 J

Since the efficiency is only 90%, we need E/0.9 to make sure we get enough.

Now you can just assume any water that passes through the turbines started at the top, so the energy it initially had was all potential
so PE = mgh
where h =150.
So we need to determin the mass of water to make the equivalent E/0.9 J of energy
mgh = E/0.9
m = E/(0.9gh)

finally, since the volume is given by Area x t, and we note water has a density of 1000kg/m^3, we note
Area x t = m/density
solve for t...

Answer 2

The potential energy of the water at that depth is, surprisingly, equal to mgh. So the amount of mass you need to move through the turbines to generate the energy required is computed from

(100 kW-h)(1000 W / kW)(3600 s / h) = (.90)(m)(9.8 m/s^2)(150 m)

1 kg of water has a volume of about .001 cubic meter. Use the mass from the previous part to find the volume, set that equal to h * surface area, and solve for h to see how much the water level drops.