Question

A body cools from a temperature $3T$ to $2T$ in 10 minutes. The room temperature is $T$. Assume that Newton’s law of cooling is applicable. The temperature of the body at the end of next 10 minutes will be

• A. $\frac{7}{4}T$
• B. $\frac{3}{2}T$
• C. $\frac{4}{3}T$
• D. $T$

Answer 1

The correct option is B $\frac{3}{2}T$

Average rate of cooling $\propto$ Average temperature of body $-$ Average temperature of surrounding
$\begin{array}{c}\frac{T_2-T_1}{\Delta t}\propto (\frac{T_1+T_2}{2}-T_0)\\ \frac{T_2-T_1}{\Delta t}=-k(\frac{T_1+T_2}{2}-T_0)\end{array}$
In case 1
$\frac{2T-3T}{10}=-k(\left(\frac{3T+2T}{2}\right)-T)=-k(2.5T-T)=-1k\left(1.5T\right)$

Caase 2

$\frac{T^{\prime }-2T}{10}=-k(\frac{T^{\prime }+2T}{2}-T)=-k\left(\frac{T^{\prime }+2T-2T}{2}\right).$

$\frac{T^{\prime }-2T}{10}=-k\left(\frac{T^{\prime }}{2}\right)$

Solving for $T^{\prime }$ we get $T^{\prime }=1.5T$