| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Moderate -0.3 This is a structured multi-part question on exponential models that guides students through standard techniques: substitution to find A, rearranging to logarithmic form, differentiation of exponentials, and solving rate equations. While it requires multiple steps and careful algebra, each part follows predictable C4-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(t=0\): \(h = A(1-1) = 0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(57 = A\left(1 - e^{-\frac{12}{4}}\right)\) | M1 | |
| \(A = \frac{57}{\left(1-e^{-3}\right)} \approx 60\) | A1 | Or 59.9… seen; \(A =\) correct expression \(\approx 60\) to 2sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h=48\): \(\frac{48}{60} = 1 - e^{-\frac{1}{4}t}\) | M1 | |
| \(\ln\left(e^{-\frac{1}{4}t}\right) = \ln\left(\frac{1}{5}\right)\) | m1 | |
| \(-\frac{1}{4}t = -\ln 5 \Rightarrow t = 4\ln 5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dh}{dt} = -\frac{1}{4}\times -60\times e^{-\frac{1}{4}t}\) | M1 | Differentiate; condone sign errors |
| \(60e^{-\frac{1}{4}t} = 60 - h \Rightarrow \frac{dh}{dt} = \frac{1}{4}(60-h)\) | m1 | Eliminate \(e^{-\frac{1}{4}t}\) |
| \(\frac{dh}{dt} = 15 - \frac{h}{4}\) | A1 | CSO, AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 8\) | B1 |
## Question 9(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $t=0$: $h = A(1-1) = 0$ | B1 | |
## Question 9(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $57 = A\left(1 - e^{-\frac{12}{4}}\right)$ | M1 | |
| $A = \frac{57}{\left(1-e^{-3}\right)} \approx 60$ | A1 | Or 59.9… seen; $A =$ correct expression $\approx 60$ to 2sf |
## Question 9(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h=48$: $\frac{48}{60} = 1 - e^{-\frac{1}{4}t}$ | M1 | |
| $\ln\left(e^{-\frac{1}{4}t}\right) = \ln\left(\frac{1}{5}\right)$ | m1 | |
| $-\frac{1}{4}t = -\ln 5 \Rightarrow t = 4\ln 5$ | A1 | |
## Question 9(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dh}{dt} = -\frac{1}{4}\times -60\times e^{-\frac{1}{4}t}$ | M1 | Differentiate; condone sign errors |
| $60e^{-\frac{1}{4}t} = 60 - h \Rightarrow \frac{dh}{dt} = \frac{1}{4}(60-h)$ | m1 | Eliminate $e^{-\frac{1}{4}t}$ |
| $\frac{dh}{dt} = 15 - \frac{h}{4}$ | A1 | CSO, AG |
## Question 9(b)(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 8$ | B1 | |9 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow.
She proposes the model $h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)$, where $A$ is a constant, for the height $h$ millimetres of the toadstool, $t$ hours after it begins to grow.
\begin{enumerate}[label=(\alph*)]
\item Use this model to:
\begin{enumerate}[label=(\roman*)]
\item find the height of the toadstool when $t = 0$;
\item show that $A = 60$, correct to two significant figures.
\end{enumerate}\item Use the model $h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)$ to:
\begin{enumerate}[label=(\roman*)]
\item show that the time $T$ hours for the toadstool to grow to a height of 48 millimetres is given by
$$T = a \ln b$$
where $a$ and $b$ are integers;
\item show that $\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }$;
\item find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.\\
(1 mark)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q9 [10]}}