Question

Assuming Newton's law of cooling to be valid, the temperature of body changes from $60^{\circ} C$ to $40^{\circ} C$ in $7\, \min$. Temperature of surrounding being $10^{\circ} C$. Find its temperature after next $7 \,\min .$

• A. $ 24^{\circ} C $
• B. $ 20^{\circ }C $
• C. $ 14^{\circ} C $
• D. $ 28^{\circ }C $

Answer 1

Answer: (D)

Let after next $7$ min, its temperature be $ \theta $ .
From Newton's law of cooling.
$ \frac{{{\theta }_{1}}-{{\theta }_{2}}}{t}\propto \left( \frac{{{\theta }_{1}}+{{\theta }_{2}}}{2}-{{\theta }_{0}} \right) $
where $ {{\theta }_{0}}= $ temperature of surrounding.
$ \therefore $ $ \frac{60-40}{7}\propto \left( \frac{60+40}{2}-10 \right) $...(i)
and $ \frac{40-\theta }{7}\propto \left( \frac{40+\theta }{2}-10 \right) $ ...(ii)
Dividing Eq. (i) by Eq. (ii), we obtain
$ \frac{20}{7}\times \frac{7}{(40-\theta )}=\frac{40}{(20+\theta )/2} $
$ \Rightarrow \frac{20}{40-\theta }=\frac{40\times 2}{20+\theta } $
$ \Rightarrow 20+\theta =160-4\theta $
$ \Rightarrow 5\theta =160-20=140 $
$ \therefore \theta =\frac{140}{5}={{28}^{\circ}}C $