17 In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of \(a\) litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of \(b\) litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After \(t\) minutes, the mixture in the vessel contains \(x\) litres of the chemical.
- Show that the proportion of chemical present in the vessel after \(t\) minutes is
$$\frac { x } { 1 + ( a - b ) t } .$$
- Hence show that \(\frac { d x } { d t } + \frac { b x } { 1 + ( a - b ) t } = a\).
- First, consider the case where \(\mathbf { b } = \mathbf { a }\).
- Solve the differential equation to find \(x\) in terms of \(a\) and \(t\).
- Given that after 1 minute the vessel contains equal amounts of water and chemical, find the rate of inflow of chemical.
- Now consider the case where \(\mathrm { b } = 2 \mathrm { a }\).
- Explain why the differential equation in part (a)(ii) is now invalid for \(\mathrm { t } \geqslant \frac { 1 } { \mathrm { a } }\).
- Find the maximum amount of chemical in the vessel.