Question

A body initially at ${80}^{\circ }C$ cools to ${64}^{\circ }C$ in $5$ min and to ${52}^{\circ }C$ in 10 min. The temperature of the surrounding is

• A. ${26}^{\circ }C$
• B. ${16}^{\circ }C$
• C. ${36}^{\circ }C$
• D. ${40}^{\circ }C$

Answer 1

Answer: (B)

${16}^{\circ }C$

According to Newton's law of cooling
$\frac{{\theta }_1-{\theta }_2}{t}=K[\frac{{\theta }_1+{\theta }_2}{2}-{\theta }_0]$
In the first case, $\frac{80-64}{5}=K[\frac{80+64}{2}-{\theta }_0]$
or $3.2=K[72-{\theta }_0]....\left(i\right)$
In the second case, $\frac{64-52}{5}=K[\frac{64+52}{2}={\theta }_0]$
or $2.4=K[58-{\theta }_0]....\left(ii\right)$
Dividing (i) by (ii), we get $\frac{3.2}{2.4}=\frac{72-{\theta }_0}{58-{\theta }_0}$
or $185.6-3.2{\theta }_0=172.8-2.4{\theta }_0$ or ${\theta }_0={16}^{\circ }C$.