8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-13_721_1227_116_322}
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\caption{Figure 3}
\end{figure}
The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be
$$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$
where \(k\) is a positive constant.
The graph of \(P\) against \(t\) is shown in Figure 3.
Use the given equation to
- find the population at the start of the study,
- find a value for the expected upper limit of the population.
Given also that the population reaches 2500 at 3 years from the start of the study,
- calculate the value of \(k\) to 3 decimal places.
Using this value for \(k\),
- find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
- Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study.