| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Logistic growth model |
| Difficulty | Standard +0.3 This is a structured logistic growth question with clear scaffolding through multiple parts. Part (a) involves routine substitution and algebraic manipulation of exponentials/logarithms. Part (b)(i) requires differentiation of a quotient (or chain rule) but the answer is given to verify. Part (b)(ii) requires finding when dN/dt is maximum, which reduces to solving a simple exponential equation. While it covers several techniques, each step is standard for C4 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties1.06g Equations with exponentials: solve a^x = b1.07l Derivative of ln(x): and related functions1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08l Interpret differential equation solutions: in context |
7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model
$$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$
for the number of rodents, $N$, in the population $t$ weeks after the start of the investigation.
Use this model to answer the following questions.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the size of the population at the start of the investigation.
\item Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
\item Find the number of weeks that it will take the population to reach 400 . Give your answer in the form $t = r \ln s$, where $r$ and $s$ are integers.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that the rate of growth, $\frac { \mathrm { d } N } { \mathrm {~d} t }$, is given by
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
\item The maximum rate of growth occurs after $T$ weeks. Find the value of $T$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q7 [13]}}