2015 AJC P2 Q1(b)

A tank initially contains \(50\) grams of salt dissolved in \(100\) litres of water. Brine that contains \(2\) grams of salt per litre of brine flows into the tank at a rate of  \(5\) litres per minute. The solution is kept thoroughly mixed and flows out from the tank at a rate of \(5\) litres per minute.

Given that the amount of salt in the tank at time t minutes is given by \(S\), show that

\(\frac{\mathrm{d}S}{\mathrm{d}t}=\frac{200-S}{20}\).

Hence find the time, in minutes, at which the concentration of salt in the tank reaches \(1\) gram per litre.[7]