If water is pumped in through the bottom of the tank in the picture, how much work is done to fill the tank:

a. To a depth of 2 feet?

b. From a depth of 4 feet to a depth of 6 feet?

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- Jun 6th 2007, 03:53 AMharryamount of work
If water is pumped in through the bottom of the tank in the picture, how much work is done to fill the tank:

a. To a depth of 2 feet?

b. From a depth of 4 feet to a depth of 6 feet? - Jun 6th 2007, 04:21 AMPterid
Here's what I make of it.

- - -

The tank is conical, clearly 6 units high, and it looks like it's roughly 4 units in 'radius' at the top.

Call the radius at height y . If and it runs linearly (as you'd expect), then in general

.

Using conservation of energy, the amount of work done to fill the tank to a height (call it ) is equal to the total*gravitational potential energy*of the water in the container.

The gravitational potential energy is found by*integrating*(calculus) over circular "slices" of water, at height and with thickness . (Let me know if you don't get this part; it's explained better with a diagram.)

The slice will have a volume

.

Its*mass*comes just from multiplying by the density of water, :

.

Finally, the potential energy of the slice is

(where I have substituted for .)

Then just integrate with respect to .

.

The answer to part a) is .

The answer to part b) is (as the tank was already filled to 4 units.)