Question

Can someone please explain what is going on here?

Answer 1

Is there some information missing or an assumption?

Explanation:

This is a potential energy question coupled with geometry and centre of mass. I'm not clear where the water is being pumped from nor what the phrase "out of the tank" is getting at. I'd guess that that the water is initially assumed to be all at the level of the point of the cone (i.e. outside the cone, as though the tank is positioned with its point on the ground and all the water is at ground level, that is, not being pumped out of a well nor is the tank of top of a tower or underground!).

In that case the minimum energy to pump water up into the tank seems to be simply the increase in potential energy of the water in going from state A (all the water at ground level) to state B(all the water is in the tank).

You therefore need to find the increase in height of the center of mass (because EnergyRequired = mgh where m is the mass of water and h is the increase in height of the center of mass). Initially the center of mass (on the assumption above) is (relative to the tip of the cone) at h=0 and then goes up to wherever the center of mass is when it is full.

So you need to find the formula for the center of mass of a cone. (Hint: it is three quarters of the way from the point to up to the base.) So h is nine meters. You then need the mass of water in the tank when it is full, which is `rho/3pi r^2H` wher `rho` is the density of water, `H=12` and `r=1.5`