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 the differential equation

\[ \frac{dN}{dt} = kN^{\frac{3}{2}} \cos(0.02t), \]

where \(k\) is a constant and \(N\) is treated as a continuous variable. It is given that when \(t = 0\), \(N = 100\).

(a) Solve the differential equation, obtaining a relation between \(N\), \(k\), and \(t\).

(b) Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).

(c) Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.