Question

An object is cooled from ${75}^{\circ }\text{C}$ to ${65}^{\circ }\text{C}$ in $2$ min in a room at ${30}^{\circ }\text{C}$. The time taken to cool another identical object from ${55}^{\circ }\text{C}$ to ${45}^{\circ }\text{C}$ in the same room, in minutes is

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is A $4$

According to Newton's law of cooling,
$-\frac{d\theta }{dt}\propto (\theta -{\theta }_0)$
$-\frac{d\theta }{dt}=K(\theta -{\theta }_0)$
where, $K$ is a constant of proportionality.
Now, $\frac{75-65}{2}=K(\frac{75+65}{2}-30)=K\times 40\Rightarrow K=\frac{1}{8}$
In case of second identical object
$\frac{55-45}{t}=K(\frac{55+45}{2}-30)$
$\frac{10}{t}=\frac{1}{8}\times 20$
$\Rightarrow t=\frac{80}{20}=4\text{min}$