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 a differential equation of the form

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

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

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.