I am working on a homework problem that presents the scenario of trying to raise water from a small reservoir of depth 8 m whose surface is 25 m below a pump that can maintain a pressure differential of 10 atm. According to the problem this is not possible, but I am charged with 1) finding the maximum height that this water could be sucked up, and 2) finding an optimal placement for the pump (instead of at the top of the hill) so that it could suck the water up. I am not given how much water is in the reservoir. I do not know where to begin. I do not really understand how the pump works (the instructor has said it does not really matter, all you need to know is that it is a "black box" that maintains this pressure differential). I would appreciate some guidance on where to begin or how to set up my equations.
There's a big difference between trying to such water up from the top, and trying to push it up from the bottom.
It we first consider trying to push up water from the bottom, then the height you can raise the water simply depends on how much pressure you can produce. A quick Google will tell you that a pressure of 1 atmosphere corresponds to a height (or depth) of about ten metres. So if your pump can produce a pressure of ten atomspheres it will be able to pump the water to about 100 metres. I say "about" because your professor probably expects you to calculate this exactly and I won't spoil their fun by just giving you the answer :-)
The reason why sucking water up is different is because water boils at reduced pressure. The pressure at the surface of the pool is obviously one atmosphere. To suck water up you have to reduce the pressure at the pump. However if you reduce the pressure too much the water starts to boil and the water won't go up any further. All you do is suck steam. The maximum height you can suck water can be calculated from it's vapour pressure and density. When the pressure has reduced to less that the water vapour pressure the water starts to boil. Again, I won't spoil your professor's fun by just giving you the answer.
This should give you some clues about how to approach the problem. If you need any more info please comment.