2025 ACJC P1 Q7

A freshly brewed cup of tea, initially at a temperature of \({85^\circ\text{C}}\), is left out to cool in a room with a room temperature of \({25^\circ\text{C}}\). After \({20}\) minutes, the temperature of the tea is \({55^\circ\text{C}}\).

It is known that the rate of change of the temperature of the tea is proportional to the difference between the temperature of the tea and the room temperature.

1. Given that \(\theta\) is the temperature of the tea \(t\) minutes after the tea is left out to cool, show that \(\theta = 25 + 60\mathrm{e}^{-kt}\), where \(k\) is a constant to be determined.[5]
2. Sketch the graph of \(\theta\) against \(t\).[2]