A freshly brewed cup of coffee has an initial temperature $95{}^\circ C$. It is placed in a room w

Question

A freshly brewed cup of coffee has an initial temperature $95{}^\circ C$. It is placed in a room where the temperature is a constant at $20{}^\circ C$. As the coffee cools down, the rate of decrease of its temperature $\theta {}^\circ C$ after time $t$ minutes is proportional to the temperature difference $\left( \theta -20 \right){}^\circ C$. It is cooling at a rate of $1{}^\circ C$ per minute when its temperature is $70{}^\circ C$. By setting up a differential equation, find the rate of cooling of the coffee when its temperature is $40{}^\circ C$ and show that $\theta =20+75{{e}^{-0.02t}}$ . Comment on whether the model can be regarded as a good model of the situation in the real world.

Accepted Answer

(b)(i) Rate at 40°C = -2/5 °C/min; θ = 20 + 75e^(-t/50) (b)(ii) θ → 20°C as t → ∞, good model

2014 NJC P1 Q11

2014 NJC P1 Q11

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