# How much work is required to pump all of the water to the top?

1. Apr 18, 2012

### IntegrateMe

Consider the vessel. It is filled to a depth of 1 foot of water. Write an integral in terms of y (the distance from the bottom) for the work required to pump all the water to the top of the vessel. Water weights 62.4 lbs/ft3.

Relevant Equations
W = ∫F dx
Pressure = density * g * depth
F = Pressure * V

Attempt

P = 62.4 * 9.8 * y = 611.52y

Based on similar triangles, I was able to get a width of (3/2)y. Thus, the volume becomes:

V = 5*(3/2)y*Δy = 7.5yΔy

So the force is F = (611.52y)(7.5y)Δy

I stopped here because I wasn't sure about my work. Any help?

2. Apr 18, 2012

### IntegrateMe

Anyone?

3. Apr 18, 2012

### HallsofIvy

Staff Emeritus
Yes, what you have done so far is correct. It would have been better to continue with what you think is correct rather than wait until someone confirms that.

And don't expect someone to respond within 30 minutes. If you continue to "bump" so soon, you may be banned from this forum.

4. Apr 18, 2012

### IntegrateMe

HallsofIvy, the answer I have is actually incorrect based on the final solution offered, which is:

$$\int_0^1 5(\frac{3}{2}y)(62.4)(2-y)dy$$

Obviously, my solution does not match this one, which is why I stopped at the force and decided to post here.

Last edited: Apr 18, 2012
5. Apr 18, 2012

### Staff: Mentor

Think about how high you need to lift (pump) each layer of water. The incremental work is the weight (a force) of a typical layer of water times the distance it has to be lifted.

6. Apr 18, 2012

### IntegrateMe

Awesome explanation Mark. Very concise and, perhaps more importantly, easy to understand. I truly appreciate it. Thank you!

7. Apr 18, 2012

### Staff: Mentor

You're welcome!