- A scientist is studying a population of mice on an island.
The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation
$$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$
- Find the number of mice in the population at the start of the study.
- Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\)
The rate of growth is a maximum after \(T\) months.
- Find, according to the model, the value of \(T\).
According to the model, the maximum number of mice on the island is \(P\).
- State the value of \(P\).