The initial temperature of a body is ${80}^0\mathrm{C}$. If its temperature falls to ${64}^0\mathrm{C} \mathrm{in} 5 \mathrm{minutes} \mathrm{and} \mathrm{in} next 10 \mathrm{minutes} \mathrm{to} {52}^0$. Find the temperature of surrounding.

According to Newton's law $\frac{{\mathrm{\theta }}_1-{\mathrm{\theta }}_2}{\mathrm{t}} = \mathrm{k}[\frac{{\mathrm{\theta }}_1+{\mathrm{\theta }}_2}{2}-{\mathrm{\theta }}_0]$

Initially, $\frac{(80-64)}{5} = \mathrm{K}(\frac{80+64}{2}-{\mathrm{\theta }}_0)$

$\Rightarrow 3.2 = \mathrm{K}[72-{\mathrm{\theta }}_0]-------\left(\mathrm{i}\right)$

Finally, $\frac{(64-52)}{10} = \mathrm{K}[\frac{64+52}{2}-{\mathrm{\theta }}_0]$

$\Rightarrow 1.2 = \mathrm{K}[58-{\mathrm{\theta }}_0]------\left(\mathrm{ii}\right)$

On solving equation (i) and (ii) ${\mathrm{\theta }}_0 = {49}^0\mathrm{C}$