This is another question i came across which i really cant get my head around. Questions like this really really stop me in my tracks i just dont know how to think about them. If somebody could give me some guidelines on answering questions like these i would be really grateful because at the moment if one of these comes up in my exam i will be in trouble!

Liquid is pouring into a large vertical circular cylinder at a constant rate of $\displaystyle 1600cm^3s^{-1}$ and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is $\displaystyle 4000cm^2$.

a)show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation: $\displaystyle \frac{dh}{dt} = 0.4-k\sqrt h$ where k is a positive constant. (3 marks)

When h = 25, water is leaking out of the hole at $\displaystyle 400cm^3s^{-1}$

b) show that k = 0.02 (1 mark)

c) Separate the variables of the differential equation:$\displaystyle \frac{dh}{dt} = 0.4-0.02\sqrt h$ to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by: $\displaystyle \int_0^{100}\frac{50}{20-\sqrt h}\,dh$ (2 marks)

Using the substitution $\displaystyle h=(20-x)^2$, or otherwise,

d) find the exact value of $\displaystyle \int_0^{100}\frac{50}{20-\sqrt h}\,dh$ (6 marks)

e) Hence find the time taken to fill the cylinder from empty to a height of 100 cm, giving your answer in minutes and seconds to the nearest second. (1 mark)