1 answer

Newton's law of cooling says that the temperature of a body changes at a rate proportional to the difference between

Question:

Newton's law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature), $$\frac{d T}{d t}=-k\left(T-T_{a}\right)$$ where $T=$ the temperature of the body $\left(^{\circ} \mathrm{C}\right), t=$ time $(\mathrm{min}), k=$ the proportionality constant (per minute), and $T_{a}=$ the ambient temperature $\left(^{\circ} \mathrm{C}\right)$. Suppose that a cup of coffee originally has a temperature of $68^{\circ} \mathrm{C}$. Use Euler's method to compute the temperature from $t=0$ to 10 min using a step size of 1 min if $T_{a}=21^{\circ} \mathrm{C}$ and $k=$ $0.017 / \mathrm{min}$.


Answers

(a) A cup of coffee has temperature 95 $ ^\circ C $ and takes 30 minutes to cool to 61 $ ^\circ C $ in a room with temperature 20 $ ^\circ C $. Use Newton's Law of Cooling (section 3.8) to show that the temperature of the coffee after $ t $ minutes is
$$ T(t) = 20 + 75 e^{-kt} $$
where $ k \approx 0.02 $.
(b) What is the average temperature of the coffee during the first half hour?

.

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