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

Answer 1

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

By using Newton's law of cooling,
$\frac{T_1-T_2}{t}=k(T_{avg}-T_s)$

Case 1:
In 10 minutes it cools from $3T$ to $2T$
$\frac{3T-2T}{10}=k(\left(\frac{3T+2T}{2}\right)-T)$
$\frac{T}{10}=k\frac{3T}{2}$
$k=\frac{1}{15}$

Case 2:
In next 10 minutes it cools from $2T$ to some temperature $T^{\prime }$
$\frac{2T-T^{\prime }}{10}=k(\left(\frac{2T+T^{\prime }}{2}\right)-T)$
$\frac{2T-T^{\prime }}{10}=\frac{1}{15}\frac{T^{\prime }}{2}$
$6T-3T^{\prime }=T^{\prime }$
$T^{\prime }=\frac{3T}{2}$