8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).

1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.