Exercise 4

A cylinder in a water purification factory has a radius of 1,25 metres, is 6 metres high and is filled with water up to a height of 4 metres. On the very bottom, the cylinder has two openings: one input and one output of water. The input is set to throughput 0,5 litres per second into the cylinder, and the output is set to throughput 0,7 litres per second out of the cylinder.


a)      Give a formula for the height h of the water in the tank, as a function of the time t. Assume that the height of the water is 4 metres at t=0, so h(0) = 4. Note that the input and output rates are constant and hence do not depend on gravity.

The factory also adds a certain mineral to the water, in order to satisfy the requirements of drinking water. This process is started by closing the output of the cylinder. The input is not closed and still has a throughput of 0,5 litres per second. The concentration of the minerals in the input is 40g/litre. When the process is started the height of the water in the cylinder is 4 metres, and the concentration is 20g/litre. A worker adds 8 kilograms of pure minerals to the cylinder.

b)      After adding the pure minerals, the worker waits 20 seconds. What will the concentration of minerals in the cylinder be after these 20 seconds? You can assume that the added pure minerals will instantaneously be homogeneously mixed throughout the tank.