2013 JJC P2 Q2

A researcher is investigating the spread of a certain disease in a town with a population of \(3000\) people. The researcher suggests that \(I\), the number of people infected by the disease at time \(t\) days satisfies the differential equation \(\frac{\mathrm{d}I}{\mathrm{d}t}=kI\left( 3000-I \right)\), where \(k\) is a positive constant.

1. Given that \(I=30\) when \(t=0\), show that \(I=\frac{3000}{1+99{{e}^{-3000kt}}}\).[6]
2. It is further observed that \(I=240\) when \(t=7\), find the time it takes for \(90\%\) of the population to be infected by the disease.[3]
3. State, in the context of this question, one assumption needed to model the spread of the disease in the town by the given differential equation.[1]