Question

A body initially at ${80}^{\circ }\text{C}$ cools to ${64}^{\circ }\text{C}$ in $5$ minutes and to ${52}^{\circ }\text{C}$ after another $5$ minutes. The temperature of the surroundings is

• A. ${15}^{\circ }\text{C}$
• B. ${16}^{\circ }\text{C}$
• C. ${20}^{\circ }\text{C}$
• D. ${25}^{\circ }\text{C}$

Answer 1

The correct option is B ${16}^{\circ }\text{C}$

Let $T_0$ is the temperature of the surroundings, $T_i$ is the initial temperature of body and $T_t$ is the temperature of body at time $t$.
According to Newton's laws of cooling we have,
${\log }_e\left(\frac{T_i-T_0}{T_t-T_0}\right)=5t$

From ${80}^{\circ }\text{C}$ to ${64}^{\circ }\text{C}$ in $5$ minutes,
${\log }_e\left(\frac{80-T_0}{64-T_0}\right)=5K...\left(1\right)$
and from ${64}^{\circ }\text{C}$ to ${52}^{\circ }\text{C}$ in next $5$ minutes,
${\log }_e\left(\frac{64-T_0}{52-T_0}\right)=5K...\left(2\right)$

Equating Eqs $\left(1\right)$ and $\left(2\right)$, we get
$\frac{80-T_0}{64-T_0}=\frac{64-T_0}{52-T_0}$
which gives $T_0={16}^{\circ }\text{C}$, which is choice (b).