15. The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).
- Interpret the constant \(A\) in the context of the question.
- Show that \(k = 0 \cdot 047\), correct to three decimal places.
- Find the size of the population when \(t = 20\).