2025 MI P1 Q12

A tank initially contains \(100\) litres of pure water. A brine solution, with a salt concentration of \(0.2\) kg per litre, is pumped into the tank continuously at a constant rate of \(5\) litres per minute. At the same time, a well-stirred mixture of the tank's contents is continuously drained from the tank at the same rate of \(5\) litres per minute. At time \(t\) minutes, the mass of salt in the tank is denoted by \(Q\) kilograms.

1. Explain briefly why the volume of the mixture in the tank remains constant.[1]
2. By considering the salt concentration of the brine solution, show that salt enters the tank at a constant rate of \(1\) kg per minute.[1]
3. Hence, by also considering the mass of salt leaving the tank, show that the rate of change of the mass of salt in the tank can be expressed as

   \(\frac{\mathrm{d}Q}{\mathrm{d}t} = 1 - \frac{Q}{20}.\)

   [2]
4. Solve the differential equation found in part (iii) to find \(Q\) in terms of \(t\).[5]
5. Determine the time taken for the salt concentration in the tank to reach \(0.1\) kg per litre.[2]
6. What happens to \(Q\) for large values of \(t\)?[1]