Question

A body cools down from \(75^\circ\text{C}\) to \(65^\circ\text{C}\) in \(10\) minutes. It will cool down from \(65^\circ\text{C}\) to \(55^\circ\text{C}\) in a time:

• A. \(10\) minutes
• B. Less than \(10\) minutes
• C. More than \(10\) minutes
• D. Less than or more than \(10\) minutes depending on its mass

Hint:

According to Newton’s law of cooling, cooling is faster when the temperature difference between the body and surroundings is larger.

Answer 1

The Correct Option is C

Step 1: State Newton's law of cooling.
According to Newton's law of cooling, the rate at which a body loses temperature is proportional to the difference between its temperature and the surrounding temperature:
\[ \frac{dT}{dt} = -k(T - T_s) \] where $k > 0$ is a cooling constant and $T_s$ is the surrounding temperature.

Step 2: Understand what the law tells us physically.
When the body is hotter (large $T - T_s$), it cools faster.
As the body cools and approaches the surroundings, the rate of cooling decreases.

Step 3: Compare the average temperature during the first interval.
During cooling from $75^\circ\text{C}$ to $65^\circ\text{C}$, the average body temperature is about $70^\circ\text{C}$.

Step 4: Compare the average temperature during the second interval.
During cooling from $65^\circ\text{C}$ to $55^\circ\text{C}$, the average body temperature is about $60^\circ\text{C}$.

Step 5: Compare the rates of cooling in the two intervals.
The second interval has a lower average temperature, meaning the temperature difference $(T - T_s)$ is smaller.
A smaller temperature difference means a slower rate of cooling.
Since the body must cool by the same $10^\circ\text{C}$ but at a slower rate, it takes more time.

Step 6: State the conclusion.
The body will cool from $65^\circ\text{C}$ to $55^\circ\text{C}$ in more than 10 minutes.
\[ \boxed{\text{More than 10 minutes}} \]