Question

A body cools from ${60}^{\circ }\text{C}$ to ${50}^{\circ }\text{C}$ in $10$ minutes. If the room temperature is ${25}^{\circ }\text{C}$ and assuming that Newton's law of cooling holds good, the temperature of the body at the end of the next $10$ minutes will be:

• A. ${38.5}^{\circ }\text{C}$
• B. ${40}^{\circ }\text{C}$
• C. ${37.5}^{\circ }\text{C}$
• D. ${42.9}^{\circ }\text{C}$

Answer 1

The correct option is D ${42.9}^{\circ }\text{C}$

From Newton's law of cooling, we know that
$\frac{{\theta }_1-{\theta }_2}{t}=k[\frac{{\theta }_1+{\theta }_2}{2}-{\theta }_0]$
In first $10$ minutes, from the data given in the question, we can say that
$\frac{60-50}{10}=k[\frac{60+50}{2}-25]$
$\Rightarrow k=\frac{1}{30}......\left(1\right)$
Let $\theta$ be the temperature at the end of next $10$ minutes.
$\Rightarrow \frac{50-\theta }{10}=k[\frac{50+\theta }{2}-25]$
Using $\left(1\right)$ in the above equation,
$\frac{50-\theta }{10}=\frac{\theta }{60}$
$\Rightarrow 7\theta =300\Rightarrow \theta \approx {42.9}^{\circ }\text{C}$
Thus, option (c) is the correct answer.