13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation
$$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$
where \(k\) is a constant.
Given that there were 50 insects at the start of the study,
- find the value of \(k\)
- use the model to find the value of \(t\) when \(N = 100\)
- Show that
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$
where \(p\) and \(q\) are integers to be found.