5. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside an oven, \(t\) minutes after the oven is switched on, is given by
$$\theta = A - 180 \mathrm { e } ^ { - k t }$$
where \(A\) and \(k\) are positive constants.
Given that the temperature inside the oven is initially \(18 ^ { \circ } \mathrm { C }\),
- find the value of \(A\).
The temperature inside the oven, 5 minutes after the oven is switched on, is \(90 ^ { \circ } \mathrm { C }\).
- Show that \(k = p \ln q\) where \(p\) and \(q\) are rational numbers to be found.
Hence find
- the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
- the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.