9. An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). Given that after one week there are 360 insects in the colony,
2. find the value of \(k\) to 3 significant figures. Given also that after two and three weeks there are 440 and 600 insects respectively,
3. comment on suitability of the modelling assumption. An alternative model assumes that $$\frac { \mathrm { d } P } { \mathrm {~d} t } = P ( 0.4 - 0.25 \cos 0.5 t )$$
4. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation.
5. Compare the suitability of the two models.