The number of insects in a population \(t\) days after the start of observations is denoted by \(N\). The variation in the number of insects is modelled by the differential equation

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

where \(k\) is a constant and \(N\) is taken to be a continuous variable. It is given that \(N = 125\) when \(t = 0\).

1. Solve the differential equation, obtaining a relation between \(N\), \(k\), and \(t\).
2. Given also that \(N = 166\) when \(t = 30\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the least value of \(N\) predicted by this model.