Question

The temperature of a body in air falls from \( 40^\circ \text{C} \) to \( 24^\circ \text{C} \) in 4 minutes. The temperature of the air is \( 16^\circ \text{C} \). The temperature of the body in the next 4 minutes will be:

• A. \( \frac{28}{3} \, ^\circ \text{C} \)
• B. \( \frac{14}{3} \, ^\circ \text{C} \)
• C. \( \frac{56}{3} \, ^\circ \text{C} \)
• D. \( \frac{42}{3} \, ^\circ \text{C} \)

Hint:

Use Newton's Law of Cooling to calculate temperature changes over time. The rate of temperature change depends on the difference between the object's temperature and the surrounding temperature.

Answer 1

The Correct Option is A

This problem involves Newton's Law of Cooling, which can be expressed as: \[ \frac{dT}{dt} = -k(T - T_{\text{air}}), \] where \( T \) is the temperature of the body, \( T_{\text{air}} \) is the ambient temperature (temperature of the air), and \( k \) is a constant. The temperature changes from \( 40^\circ \text{C} \) to \( 24^\circ \text{C} \) in 4 minutes. Using Newton's Law of Cooling, we can compute the constant \( k \) and then apply it to determine the temperature change in the next 4 minutes. Based on the given information and applying the necessary calculations, the temperature after 4 more minutes is: \[ T = 24 - \left( 24 - 16 \right) \times \left( \frac{4}{4 + 4} \right) = \frac{28}{3} \, ^\circ \text{C}. \] Final Answer: \( \frac{28}{3} \, ^\circ \text{C} \).