Work required to pump water out of a conical tank

[jollyrancher]

Homework Statement

Find the work required to empty a 10m high conical tank with a radius at the top of 4m by pumping the water out the top of the tank. The water level is 2m below the top of the tank.

Homework Equations

$\pi r^2$

Similar triangles

The Attempt at a Solution

General formula for the radius:

$\frac{4}{10} = \frac{r}{(10-x)}$ where x = distance from the top of the tank.

$r = \frac{2}{5}(10-x)$

Therefore, the volume of any given slice $= \pi(\frac{2}{5}(10-x))^2\Delta x$

This is where I get a bit confused. What do I need to multiply my volume formula by when I integrate from x = 2 to x = 10?

Also, after I have multiplied my volume by the whatever I need to (distance?) and integrated between 2 and 10, have I pumped the water out the top of the barrel or only to the top of the water level? Do I need to add +2m to something somewhere?

Thanks.

 

[BruceW]

Its actually a tricky question, which is maybe why there are no replies yet. You have the right answer for the radius of the cone as a function of x. (As long as you remember the units you are using, since strictly the 10 should be 10m).

And you have correctly given the volume of a small slice. And you are right, you need to multiply by x before you do the integration. (Or maybe you should multiply by -x, since your coordinate system is upside-down). Also, you need to multiply by the density and g to get the correct units for energy.

So once you've done this, you will have the GPE of the water before it is pumped. You can then work out what the GPE would be if all the water was at the top of the barrel (easy, since it is all at the same height). Then subtract one GPE by the other GPE to get the change in GPE.