Q.34 A body cools according to Newton's law from 100°C to 60°C in 20 minutes. The temperature of the surrounding being 20°C. How long will it take to cool down to 40°C?
Best answer
Answer: 40 minutes (from the start), i.e. 20 more minutes after reaching 60°C.
Let \(\theta\) be the temperature at time \(t\) minutes. By Newton's law of cooling, \(\frac{d\theta}{dt} = -k(\theta - 20)\), \(k \gt 0\).
\(\int\frac{d\theta}{\theta - 20} = -k\int dt \Rightarrow \log(\theta - 20) = -kt + c\).
At \(t = 0\), \(\theta = 100\): \(c = \log 80\). So \(\log\frac{\theta - 20}{80} = -kt\).
At \(t = 20\), \(\theta = 60\): \(\log\frac{40}{80} = -20k\), so \(k = \frac{1}{20}\log 2\).
For \(\theta = 40\): \(\log\frac{20}{80} = -kt \Rightarrow \log 4 = \frac{t}{20}\log 2 \Rightarrow 2\log 2 = \frac{t}{20}\log 2 \Rightarrow t = 40\).
The body takes 40 minutes to cool from 100°C to 40°C.
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