11.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-26_462_586_148_593}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

The heart rate of a horse is being monitored.\\
The heart rate $H$, measured in beats per minute (bpm), is modelled by the equation

$$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$

where $t$ minutes is the time after monitoring began.\\
Figure 4 is a sketch of $H$ against $t$.

\section*{Use the equation of the model to answer parts (a) to (e).}
\begin{enumerate}[label=(\alph*)]
\item State the initial heart rate of the horse.

In the long term, the heart rate of the horse approaches $L \mathrm { bpm }$.
\item State the value of $L$.

The heart rate of the horse reaches its maximum value after $T$ minutes.
\item Find the value of $T$, giving your answer to 3 decimal places.\\
(Solutions based entirely on calculator technology are not acceptable.)

The heart rate of the horse is 37 bpm after $M$ minutes.
\item Show that $M$ is a solution of the equation

$$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$

Using the iteration formula

$$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
\item \begin{enumerate}[label=(\roman*)]
\item find, to 4 decimal places, the value of $t _ { 2 }$
\item find, to 4 decimal places, the value of $M$
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q11 [10]}}