11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-26_462_586_148_593}
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\caption{Figure 4}
\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).}
- State the initial heart rate of the horse.
In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
- State the value of \(L\).
The heart rate of the horse reaches its maximum value after \(T\) minutes.
- 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. - 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
$$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
- find, to 4 decimal places, the value of \(t _ { 2 }\)
- find, to 4 decimal places, the value of \(M\)