10. A population of insects is being studied. The number of insects, \(N\), in the population, is modelled by the equation
$$N = \frac { 300 } { 3 + 17 \mathrm { e } ^ { - 0.2 t } } \quad t \in \mathbb { R } , t \geqslant 0$$
where \(t\) is the time, in weeks, from the start of the study.
Using the model,
- find the number of insects at the start of the study,
- find the number of insects when \(t = 10\),
- find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.)
- Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when \(t = 5\). Give your answer to the nearest whole number.