Question

An ordinary body cools from \(4θ\) to \(3θ\) in 't' minutes. The temperature of the body after next 't' minutes is (Assume Newton's law of cooling and room temperature as \(θ\))

• A. \(\frac{2θ}{3}\)
• B. \(\frac{7θ}{3}\)
• C. \(\frac{5θ}{3}\)
• D. \(\frac{8θ}{3}\)

Hint:

The excess temperature over the room falls by the same factor each interval.

Answer 1

The Correct Option is B

Step 1: Average-temperature form:
Use $\dfrac{T_1-T_2}{t}=k\left(\dfrac{T_1+T_2}2-T_0\right)$ for each interval.

Step 2: First interval:
$\dfrac{4\theta-3\theta}{t}=k\left(\dfrac{7\theta}{2}-\theta\right)=\dfrac{5k\theta}2$, so $kt=\dfrac25$.

Step 3: Second interval:
$3\theta-T=kt\left(\dfrac{3\theta+T}{2}-\theta\right)=\dfrac25\cdot\dfrac{\theta+T}2=\dfrac{\theta+T}5$.

Step 4: Solve:
$15\theta-5T=\theta+T$, so $T=\dfrac{14\theta}6=\dfrac{7\theta}3$. Option (B).

Final Answer:
Both the exponential and the averaged form give 7 theta / 3. \[ \boxed{B} \]