Differential Equations - Mixing and Concentration Models

A swimming pool contains \(375000\) litres of pure water. Water containing \(s\) milligrams of free chlorine per litre flows into the pool at a rate of \(10\) litres per minute. The pool is also draining at a rate of \(10\) litres per minute, such that the volume of the water in the pool remains constant.

1. Show that the rate at which the mass of the free chlorine, \(x\) grams, is changing in the pool over time, \(t\) minutes, can be modelled by the differential equation
   \[\frac{\mathrm{d}x}{\mathrm{d}t} = \frac{375s - x}{37500},\]State a necessary assumption for the above model to be valid.[3]
2. Given that no free chlorine is present in the pool initially, find \(x\) in terms of \(t\) and \(s\).[4]
3. Find, in terms of \(s\), the mass of free chlorine present in the pool after one hour.[1]
4. Sketch the graph of \(x\) against \(t\) which is relevant to the context, labelling the point(s) where the curve crosses the axes and the equation(s) of the asymptote(s). Hence determine the mass of free chlorine present in the pool after a long period in terms of \(s\).[3]

To prevent health complications, the recommended safe level of free chlorine to be used is between \(375\) grams and \(1125\) grams in a pool which contains \(375000\) litres of water. Find the range of values of \(s\) that should be used.[2]