8.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-26_663_1454_210_242}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\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.
\begin{enumerate}[label=(\alph*)]
\item Calculate the number of rabbits that were introduced onto the island.
\item Find $\frac { \mathrm { d } P } { \mathrm {~d} t }$

The number of rabbits initially increases, reaching a maximum value $P _ { T }$ when $t = T$
\item Using your answer from part (b), calculate
\begin{enumerate}[label=(\roman*)]
\item the value of $T$ to 2 decimal places,
\item 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.
\end{enumerate}\item Use the model to state the maximum value of $k$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel C3 2017 Q8 [9]}}