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A body cools in a surrounding which is at a constant temperature of ${\theta _0}$. Assume that it obeys Newton's law of cooling. Its temperature $\theta $ is plotted against time $t$ . Tangents are drawn to the curve at the points $P(\theta = {\theta _1})$ and $Q(\theta = {\theta _2})$. These tangents meet the time axis at angles of ${\varphi _2}$and ${\varphi _1}$, as shown
$\frac{{\tan \,{\varphi _2}}}{{\tan \,{\varphi _1}}} = \frac{{{\theta _1} - {\theta _0}}}{{{\theta _2} - {\theta _0}}}$
$\frac{{\tan \,{\varphi _2}}}{{\tan \,{\varphi _1}}} = \frac{{{\theta _2} - {\theta _0}}}{{{\theta _1} - {\theta _0}}}$
$\frac{{\tan \,{\varphi _1}}}{{\tan \,{\varphi _2}}} = \frac{{{\theta _1}}}{{{\theta _2}}}$
$\frac{{\tan \,{\varphi _1}}}{{\tan \,{\varphi _2}}} = \frac{{{\theta _2}}}{{{\theta _1}}}$
Solution
(b) For ?-t plot, rate of cooling $ = \frac{{d\theta }}{{dt}} = $slope of the curve.
At P,$\frac{{d\theta }}{{dt}} = \tan {\varphi _2} = k({\theta _2} – {\theta _0})$, where k = constant.
At Q$\frac{{d\theta }}{{dt}} = \tan {\varphi _1} = k({\theta _1} – {\theta _0})$
$ \Rightarrow \,\,\,\frac{{\tan {\varphi _2}}}{{\tan {\varphi _1}}} = \frac{{{\theta _2} – {\theta _0}}}{{{\theta _1} – {\theta _0}}}$
