8 The temperature \(\theta ^ { \circ } \mathrm { C }\) of an oven \(t\) minutes after it is switched on can be modelled by the equation $$\theta = 20 \left( 11 - 10 \mathrm { e } ^ { - k t } \right)$$ where \(k\) is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is \(T ^ { \circ } \mathrm { C }\)
The temperature predicted by the model is shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-12_750_1319_870_424} 8

1. Find the room temperature.
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2. Find the value of \(T\)
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   Question 8 continues on the next page 8
3. The oven reaches a temperature of \(86 ^ { \circ } \mathrm { C }\) one minute after it is switched on. 8
4. 1. Find the value of \(k\).
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5. (ii) Find the time it takes for the temperature of the oven to be within \(1 ^ { \circ } \mathrm { C }\) of its maximum.
   \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-15_2493_1759_173_119} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-16_805_869_459_651}
   \end{figure} The centre of the circle is \(P\) and the circle intersects the \(y\)-axis at \(Q\) as shown in Figure 1. The equation of the circle is $$x ^ { 2 } + y ^ { 2 } = 12 y - 8 x - 27$$