2025 RI P1 Q9

Decontamination is a water treatment method to remove a certain undesirable chemical from contaminated water. As the liquid agent is continuously added to the contaminated water in a large tank, the undesirable chemical in the water is removed gradually. The rate of decrease of the concentration of the undesirable chemical is proportional to the product of its current concentration and the rate at which the liquid agent flows into the tank. At time \(t\) minutes after the start of the treatment process, the concentration of the undesirable chemical is \(C\) mg/L and the liquid agent flows into the tank at a rate of \(\frac{1}{1 + t^2}\) L/min. It is known that the initial concentration of the undesirable chemical in the tank is \(50\) mg/L and after \(10\) minutes, its concentration is measured to be \(20\) mg/L.

1. By setting up and solving a differential equation relating \(C\) and \(t\), show that \(C = 50\mathrm{e}^{q \tan^{-1} t}\), giving the value of \(q\) correct to \(5\) decimal places.[7]
2. Determine the value of \(C\) when \(t = 50\).[1]
3. Sketch the graph of \(C\) against \(t\) and state what happens to \(C\) in the tank for large values of \(t\).[3]