| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Moderate -0.8 This is a straightforward application question testing basic exponential function skills: substitution (part a), solving exponential equations using logarithms (part b), differentiation and evaluation (part c), and interpreting the horizontal asymptote (part d). All parts follow standard C3 procedures with no problem-solving insight required, making it easier than average but not trivial due to the multi-step nature and need for calculator work. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks |
|---|---|
| \(425°C\) | B1 (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(300 = 400e^{-0.05t} + 25 \Rightarrow 400e^{-0.05t} = 275\) | M1 | Sub. \(T=300\) and attempt to rearrange to \(e^{-0.05t} = a\), where \(a \in \mathbb{Q}\) |
| \(e^{-0.05t} = \frac{275}{400}\) | A1 | |
| Correct application of logs | M1 | |
| \(t = 7.49\) | A1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dT}{dt} = -20e^{-0.05t}\) | M1A1 | M1 for \(ke^{-0.05t}\) |
| At \(t=50\), rate of decrease \(= (\pm)1.64°C/\text{min}\) | A1 (3) |
| Answer | Marks |
|---|---|
| \(T > 25\), (since \(e^{-0.05t} \to 0\) as \(t \to \infty\)) | B1 (1) |
# Question 4:
## Part (a)
$425°C$ | B1 (1) |
## Part (b)
$300 = 400e^{-0.05t} + 25 \Rightarrow 400e^{-0.05t} = 275$ | M1 | Sub. $T=300$ and attempt to rearrange to $e^{-0.05t} = a$, where $a \in \mathbb{Q}$
$e^{-0.05t} = \frac{275}{400}$ | A1 |
Correct application of logs | M1 |
$t = 7.49$ | A1 (4) |
## Part (c)
$\frac{dT}{dt} = -20e^{-0.05t}$ | M1A1 | M1 for $ke^{-0.05t}$
At $t=50$, rate of decrease $= (\pm)1.64°C/\text{min}$ | A1 (3) |
## Part (d)
$T > 25$, (since $e^{-0.05t} \to 0$ as $t \to \infty$) | B1 (1) |
---\begin{enumerate}
\item A heated metal ball is dropped into a liquid. As the ball cools, its temperature, $T ^ { \circ } \mathrm { C }$, $t$ minutes after it enters the liquid, is given by
\end{enumerate}
$$T = 400 \mathrm { e } ^ { - 0.05 t } + 25 , \quad t \geqslant 0$$
(a) Find the temperature of the ball as it enters the liquid.\\
(b) Find the value of $t$ for which $T = 300$, giving your answer to 3 significant figures.\\
(c) Find the rate at which the temperature of the ball is decreasing at the instant when $t = 50$. Give your answer in ${ } ^ { \circ } \mathrm { C }$ per minute to 3 significant figures.\\
(d) From the equation for temperature $T$ in terms of $t$, given above, explain why the temperature of the ball can never fall to $20 ^ { \circ } \mathrm { C }$.\\
\hfill \mbox{\textit{Edexcel C3 2006 Q4 [9]}}