8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-26_663_1454_210_242}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The number of rabbits on an island is modelled by the equation
$$P = \frac { 100 \mathrm { e } ^ { - 0.1 t } } { 1 + 3 \mathrm { e } ^ { - 0.9 t } } + 40 , \quad t \in \mathbb { R } , t \geqslant 0$$
where \(P\) is the number of rabbits, \(t\) years after they were introduced onto the island.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 3.
- Calculate the number of rabbits that were introduced onto the island.
- Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
The number of rabbits initially increases, reaching a maximum value \(P _ { T }\) when \(t = T\)
- Using your answer from part (b), calculate
- the value of \(T\) to 2 decimal places,
- the value of \(P _ { T }\) to the nearest integer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
For \(t > T\), the number of rabbits decreases, as shown in Figure 3, but never falls below \(k\), where \(k\) is a positive constant.
- Use the model to state the maximum value of \(k\).