2012 AJC P2 Q4 (b)

A new drug for the treatment of diabetes is administered to a patient at a constant rate of \(R\) mg per day. The rate at which the drug is lost form the patient’s body is proportional to the square of the amount \(x\)(mg) of the drug present in his body at time \(t\)(days).

1. If the amount of drug in the patient remains constant at the instant when it is \(2R\)(mg), show that \(\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{4{{R}^{2}}-{{x}^{2}}}{4R}\). Given that , when \(x=0\) when \(t=0\), find \(x\), in terms of \(R\) and \(t\).[5]
2. Explain the significance of this result in the long run.[1]