9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-17_574_1333_260_303}
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\caption{Figure 2}
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The population of a species of animal is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be
$$P = \frac { 9000 \mathrm { e } ^ { k t } } { 3 \mathrm { e } ^ { k t } + 7 } , \quad t \geqslant 0$$
where \(k\) is a positive constant.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 2 .
Use the given equation to
- find the population at the start of the study,
- find the value for the upper limit of the population.
Given that \(P = 2500\) when \(t = 4\)
- calculate the value of \(k\), giving your answer to 3 decimal places.
Using this value for \(k\),
- find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is increasing when \(t = 10\)
Give your answer to the nearest integer.