Question

A bucket full of hot water cools from ${75}^{\circ }\text{C}$ to ${70}^{\circ }\text{C}$ in time $T_1$, from ${70}^{\circ }\text{C}$ to ${65}^{\circ }\text{C}$ in time $T_2$ and from ${65}^{\circ }\text{C}$ to ${60}^{\circ }\text{C}$ in time $T_3$. Then:

• A. $T_1=T_2=T_3$
• B. $T_1>T_2>T_3$
• C. $T_1<T_2<T_3$
• D. $T_1>T_2<T_3$

Answer 1

The correct option is C $T_1<T_2<T_3$

From Newton's law of cooling, we know that,
$\frac{{\theta }_1-{\theta }_2}{T}\propto (\frac{{\theta }_1+{\theta }_2}{2}-{\theta }_0)$
where $T$ is the time taken for cooling.
From the data given in the question, we find $\frac{{\theta }_1+{\theta }_2}{2}$ in every case.
$\because {\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}_1>{\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}_2>{\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}_3$
[i.e ${72.5}^{\circ }\text{C}>{67.5}^{\circ }\text{C}>{62.5}^{\circ }\text{C}$]
and ${\theta }_1-{\theta }_2=5^{\circ }\text{C}$ is the same in all cases,
we can say that, $T_1<T_2<T_3$.
Hence, option (c) is the correct answer.