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
$\frac{20}{7}\times \frac{7}{(40-\theta )}=\frac{40}{(20+\theta )/2}$
$\Rightarrow \frac{20}{40-\theta }=\frac{40\times 2}{20+\theta }$
$\Rightarrow 20+\theta =160-4\theta$
$\Rightarrow 5\theta =160-20=140$
$\therefore \theta =\frac{140}{5}={28}^{\circ }C$