| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Finding x from given y value |
| Difficulty | Moderate -0.3 This is a standard exponential modelling question requiring routine application of logarithms to find k, then straightforward substitution and differentiation. All three parts follow predictable patterns typical of C3 exponential growth questions, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks |
|---|---|
| \(k = \frac{1}{3}\ln 9 = 0.732\) (3sf) | M1 M1 A1 |
| (b) \(4000 = 2000e^{0.7324t}\) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\therefore\) doubles in 57 minutes (nearest minute) | M1 A1 | |
| (c) \(N = 2000e^{0.7324t}\), \(\frac{dN}{dt} = 0.7324 \times 2000e^{0.7324t} = 1465e^{0.7324t}\) | M1 A1 | |
| When \(t = 3\), \(\frac{dN}{dt} = 13200 \therefore\) increasing at rate of 13200 per hour (3sf) | A1 | (10) |
**(a)** $t = 3, N = 18000 \Rightarrow 18000 = 2000e^{3k}$, $e^{3k} = 9$
$k = \frac{1}{3}\ln 9 = 0.732$ (3sf) | M1 M1 A1
**(b)** $4000 = 2000e^{0.7324t}$ | M1
$t = \frac{\ln 2}{0.7324} \ln 2 = 0.9464$ hours
$\therefore$ doubles in 57 minutes (nearest minute) | M1 A1
**(c)** $N = 2000e^{0.7324t}$, $\frac{dN}{dt} = 0.7324 \times 2000e^{0.7324t} = 1465e^{0.7324t}$ | M1 A1
When $t = 3$, $\frac{dN}{dt} = 13200 \therefore$ increasing at rate of 13200 per hour (3sf) | A1 | **(10)**5. The number of bacteria present in a culture at time $t$ hours is modelled by the continuous variable $N$ and the relationship
$$N = 2000 \mathrm { e } ^ { k t } ,$$
where $k$ is a constant.
Given that when $t = 3 , N = 18000$, find
\begin{enumerate}[label=(\alph*)]
\item the value of $k$ to 3 significant figures,
\item how long it takes for the number of bacteria present to double, giving your answer to the nearest minute,
\item the rate at which the number of bacteria is increasing when $t = 3$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q5 [10]}}