A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, $N$, in the first population is modelled by the equation
$$N = Ae^{kt} \quad t \geq 0$$
where $A$ and $k$ are positive constants and $t$ is the time in hours from the start of the study.

Given that
• there were 1000 bacteria in this population at the start of the study
• it took exactly 5 hours from the start of the study for this population to double

\begin{enumerate}[label=(\alph*)]
\item find a complete equation for the model.
[4]

\item Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.
[2]
\end{enumerate}

The number of bacteria, $M$, in the second population is modelled by the equation
$$M = 500e^{1.4t} \quad t \geq 0$$
where $k$ has the value found in part (a) and $t$ is the time in hours from the start of the study.

Given that $T$ hours after the start of the study, the number of bacteria in the two different populations was the same,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the value of $T$.
[3]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q6 [9]}}