June 2024 Paper 3 Q8

8 The temperature \(\theta\) °C of an oven \(t\) minutes after it is switched on can be modelled by the equation

\[\theta = 20\left(11 - 10\mathrm{e}^{-kt}\right)\]

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.

(a) Find the room temperature. [2 marks]

(b) Find the value of \(T\) [2 marks]

(i) Find the value of \(k\). [2 marks]

(ii) Find the time it takes for the temperature of the oven to be within 1 °C of its maximum. [2 marks]

Mark scheme (a)

Scheme	Marks	AO
Substitutes \(t = 0\) into the model PI by 20	M1	3.4
Obtains 20 °C Must have units	A1	3.2a
	(2)

\[\theta = 20\left(11 - 10\mathrm{e}^0\right)\]\[= 20\]

Room temperature = 20 °C

Mark scheme (b)

Scheme	Marks	AO
Replaces \(\mathrm{e}^{-kt}\) with 0 or substitutes any positive value for \(kt\) PI by 220	M1	3.4
Obtains 220	A1	3.4
	(2)

For large values of \(t\), \(\mathrm{e}^{-kt} \to 0\)

\[T = 20(11 - 10 \times 0)\]

Hence \(T = 220\)

Mark scheme (c)

Scheme	Marks	AO
(i) Forms the equation \(86 = 20\left(11 - 10\mathrm{e}^{-k}\right)\) PI by correct answer	M1	3.4
Obtains AWFW [0.4, 0.4005] or \(-\ln 0.67\) OE	A1	3.3
	(2)
(ii) Uses their \(T\) from part 8(b) and their \(k\) from part 8(c)(i) correctly to form the equation \(T - 1 = 20\left(11 - 10\mathrm{e}^{-kt}\right)\) PI by correct answer Condone use of inequality sign	M1	3.4
Obtains AWFW [13.2, 13.25] mins or AWFW [13m 12s, 13m 15s] or 13 mins Condone missing units or \(t \gt 13.2\) or \(t \geqslant 13.2\) ISW	A1	1.1b
	(2)
	(8 marks)

(i)

\[86 = 20\left(11 - 10\mathrm{e}^{-k}\right)\]\[k = 0.4\]

(ii)

\[220 - 1 = 20\left(11 - 10\mathrm{e}^{-0.4t}\right)\]\[t = 13.2\text{ minutes}\]