Moderate -0.3 This is a straightforward exponential modeling question requiring standard techniques: substituting t=0 for initial conditions, finding limits as t→∞, and solving exponential equations using logarithms. All parts follow predictable patterns with clear signposting and no novel problem-solving required, making it slightly easier than average.
The temperature \(\theta\) °C of an oven \(t\) minutes after it is switched on can be modelled by the equation
$$\theta = 20(11 - 10e^{-kt})$$
where \(k\) is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is \(T\) °C
The temperature predicted by the model is shown in the graph below.
\includegraphics{figure_8}
\begin{enumerate}[label=(\alph*)]
\item Find the room temperature.
[2 marks]
\item Find the value of \(T\)
[2 marks]
\item The oven reaches a temperature of 86 °C one minute after it is switched on.
Find the value of \(k\).
[2 marks]
Find the time it takes for the temperature of the oven to be within 1°C of its maximum.
[2 marks]
Question 8:
--- 8(a) ---
8(a) | Substitutes t = 0 into the model
PI by 20 | 3.4 | M1 | ( )
e = 2 0 1 1 − 1 0 0
= 2 0
Room temperature = 20°C
Obtains 20°C
Must have units | 3.2a | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 8(b) ---
8(b) | Replaces e − k t with 0 or
substitutes any positive value
for kt
PI by 220 | 3.4 | M1 | For large values of t, e − k t → 0
T = 2 0 ( 1 1 − 1 0 0 )
Hence T = 220
Obtains 220 | 3.4 | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 8(c)(i) ---
8(c)(i) | Forms the equation
86=20 ( 11−10e −k )
PI by correct answer | 3.4 | M1 | 86=20 ( 11−10e −k )
k = 0.4
Obtains AWFW [0.4, 0.4005] or
− l n 0 .6 7 OE | 3.3 | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 8(c)(ii) ---
8(c)(ii) | Uses their T from part 8(b) and
their k from part 8(c)(i) correctly
to form the equation
T − 1 = 2 0 ( 1 1 − 1 0 e − k t )
PI by correct answer
Condone use of inequality sign | 3.4 | M1 | 2 2 0 − 1 = 2 0 ( 1 1 − 1 0 e − 0 .4 t )
t = 13.2 minutes
Obtains
AWFW [13.2, 13.25] mins
or
AWFW [13m 12s, 13m 15s]
or
13 mins
Condone missing units or
t > 13.2 or t ≥ 13.2
ISW | 1.1b | A1
Subtotal | 2
Question 8 Total | 8
Q | Marking instructions | AO | Marks | Typical solution
The temperature $\theta$ °C of an oven $t$ minutes after it is switched on can be modelled by the equation
$$\theta = 20(11 - 10e^{-kt})$$
where $k$ is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is $T$ °C
The temperature predicted by the model is shown in the graph below.
\includegraphics{figure_8}
\begin{enumerate}[label=(\alph*)]
\item Find the room temperature.
[2 marks]
\item Find the value of $T$
[2 marks]
\item The oven reaches a temperature of 86 °C one minute after it is switched on.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $k$.
[2 marks]
\item Find the time it takes for the temperature of the oven to be within 1°C of its maximum.
[2 marks]
\end{enumerate}
</end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2024 Q8 [8]}}