2024 VJC H1 Q6

The population, \(P\) thousand, of a certain species of bats in a cave is modelled by

\[P = a - b\mathrm{e}^{-ct},\]

where \(t\) is the time in years and \(a\), \(b\) and \(c\) are constants. When \(t = 0\) the population is \(7000\) and the rate of increase of population is \(540\) per year. The population approaches \(25\,000\) in the long run.

1. Find the values of \(a\), \(b\) and \(c\).[4]
2. Sketch the graph of \(P\) against \(t\).[1]

The population, \(Q\) thousand, of another species of bats in another cave is modelled by

\[Q = 3\sqrt{t} + \frac{150}{\sqrt{t^3}}, t > 0.\]

3. Using differentiation, find the \(t\)-coordinate of the stationary point on the graph of \(Q\) against \(t\) and determine the nature of the stationary point.[4]
4. Find the duration of time, in years, for which the population \(P\) is more than the population \(Q\).[2]