Question

Hot water in a vessel kept in a room, cools from \(70^\circ C\) to \(65^\circ C\) in \(t_1\) minutes, from \(65^\circ C\) to \(60^\circ C\) in \(t_2\) minutes and from \(60^\circ C\) to \(55^\circ C\) in \(t_3\) minutes. Then

• A. \(t_1 t_3\)
• B. \(t_1 = t_2 = t_3\)
• C. \(t_1 > t_2 > t_3\)
• D. \(t_1 > t_2 = t_3\)
• E. \(t_1 < t_2 < t_3\)

Hint:

Cooling slows down as object approaches room temperature.

Answer 1

Answer: (E)

Concept: Newton’s law of cooling: \[ \frac{dT}{dt} \propto (T - T_{\text{room}}) \] Rate of cooling depends on temperature difference with surroundings.

Step 1: Initial condition. At higher temperature, difference \((T - T_{room})\) is large → cooling faster.

Step 2: As temperature decreases. Temperature difference decreases → cooling becomes slower.

Step 3: Interpret intervals.
• \(70 \to 65\): largest temperature difference → fastest → least time
• \(65 \to 60\): moderate difference → more time
• \(60 \to 55\): smallest difference → slowest → maximum time

Step 4: Conclusion. \[ t_1 < t_2 < t_3 \]