Question

A body cools from $60^\circ\text{C}$ to $40^\circ\text{C}$ in 6 minutes. After next 6 minutes its temperature will be (Temperature of the surroundings is $10^\circ\text{C}$ )}

• A. $24^\circ\text{C}$
• B. $28^\circ\text{C}$
• C. $18^\circ\text{C}$
• D. $32^\circ\text{C}$

Hint:

Cooling rates slow down as the body gets closer to the surrounding temperature. The drop in the second interval will always be less than the first.

Answer 1

The Correct Option is B

Step 1: Concept
Use Newton's Law of Cooling: $\frac{\theta_1 - \theta_2}{t} = K \left( \frac{\theta_1 + \theta_2}{2} - \theta_s \right)$.

Step 2: Meaning
First, find $K$ using the first 6 minutes. $\theta_s = 10^\circ\text{C}$.

Step 3: Analysis
Case 1: $\frac{60 - 40}{6} = K \left( \frac{60 + 40}{2} - 10 \right) \implies \frac{20}{6} = K(40) \implies K = \frac{1}{12}$.
Case 2: Let final temperature be $\theta$. $\frac{40 - \theta}{6} = \frac{1}{12} \left( \frac{40 + \theta}{2} - 10 \right)$.
$40 - \theta = \frac{1}{2} \left( \frac{40 + \theta - 20}{2} \right) = \frac{20 + \theta}{4}$.
$160 - 4\theta = 20 + \theta \implies 5\theta = 140 \implies \theta = 28^\circ\text{C}$.

Step 4: Conclusion
The temperature after next 6 minutes will be $28^\circ\text{C}$. Final Answer: (B)