9.
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The diagram shows a graph of the temperature of a room, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes.
The temperature is controlled by a thermostat such that when the temperature falls to \(12 ^ { \circ } \mathrm { C }\), a heater is turned on until the temperature reaches \(18 ^ { \circ } \mathrm { C }\). The room then cools until the temperature again falls to \(12 ^ { \circ } \mathrm { C }\).
For \(t\) in the interval \(10 \leq t \leq 60 , T\) is given by
$$T = 5 + A \mathrm { e } ^ { - k t } ,$$
where \(A\) and \(k\) are constants.
Given that \(T = 18\) when \(t = 10\) and that \(T = 12\) when \(t = 60\),
- show that \(k = 0.0124\) to 3 significant figures and find the value of \(A\),
- find the rate at which the temperature of the room is decreasing when \(t = 20\).
The temperature again reaches \(18 ^ { \circ } \mathrm { C }\) when \(t = 70\) and the graph for \(70 \leq t \leq 120\) is a translation of the graph for \(10 \leq t \leq 60\).
- Find the value of the constant \(B\) such that for \(70 \leq t \leq 120\)
$$T = 5 + B \mathrm { e } ^ { - k t }$$