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 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 .

a)show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation: where k is a positive constant. (3 marks)

When h = 25, water is leaking out of the hole at

b) show that k = 0.02 (1 mark)

c) Separate the variables of the differential equation: to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by: (2 marks)

Using the substitution , or otherwise,

d) find the exact value of (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)