4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula.
\(P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }\)
- Write down the maximum size of the population.
- Write down the difference between the largest and smallest values of \(P\).
- Without giving any numerical values, describe briefly the behaviour of the population over time.
- Find the time taken for the population to return to its initial size for the first time.
- Determine the time on the second occasion when \(P = 4500\).
A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000 .
- Suggest a change to the model that will take account of this observation.