The hardest thing about rates of change questions is deriving the equation. So let's have a think about it.
is the rate of change of the height of the liquid with respect to time. So what is causing the height to change? Well the liquid being poured in and the liquid leaking out. So let's write
Rate of change of height = rate of increase in height - rate of decrease in height.
Or another way, the combined rates of change, but remembering that the liquid leaking out gives a negative rate of change.
Well we are told that the rate of decrease is proportional to the square root of the height at that given time, so this is simply
Oh I forgot to add that if k<0 then we would be adding the rates of change and the liquid leaking out would cause the height to increase faster, which is obviously false, so we must have that k>0
Well for the rate of increase in height it's a little harder, but not much. The volume is increasing by 1600cm^3/s, the area of the cross section of the cylinder is 4000cm^2. Well the height is the volume over the cross section, so after a time t, this would be (remember we're assuming there is no liquid leaking out for this bit)
So the rate of change of height is this expression, differentiated wrt t, which is 0.4
So combining these two values gives us the overall expression for rate of change of height, that is
as required. Well hope that gets you up and running, let us know if you're having any trouble with the rest of the quesion