A freshly brewed cup of tea, initially at a temperature of $85$$^\circ C$, is left out to cool i

Question

A freshly brewed cup of tea, initially at a temperature of $85$$^\circ C$, is left out to cool in a room with a room temperature of $25$$^\circ C$. After $20$ minutes, the temperature of the tea is $55$$^\circ 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. 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] Sketch the graph of $\theta$ against $t$.[2]

Accepted Answer

(a) $\theta=25+60e^{-(\ln2)t/20}$, where $k=\dfrac{\ln2}{20}\approx0.0347$ (b) Sketch shown, with horizontal asymptote $\theta=25$

2025 ACJC P1 Q7

2025 ACJC P1 Q7

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