2015 VJC P1 Q7

A tank initially contains \(400\) litres of solution with \(100\) kg of salt dissolved in it.  A solution containing \(0.125\) kg of salt per litre flows into the tank at a rate of \(12\) litres per minute and the solution flows out at the same rate. You should assume that the inflow is instantaneously and thoroughly mixed with the contents of the tank. If the amount of salt in the tank is \(q\) kg at the end of \(t\) minutes, show that

\(\frac{\mathrm{d}q}{\mathrm{d}t}\ =\ 1.5-0.03q\).[2]

Find the time taken for the concentration of salt in the tank to reach 0.16 kg per litre.[5]

State what happens to \(q\) for large values of \(t\). Sketch a graph of \(q\) against \(t\).

(Concentration of salt = the amount of salt per unit volume of solution in the tank.)