Question

A body cools from a temperature \(3θ\) to \(2θ\) in 10 minute. The room temperature is \(θ\). The temperature of the body at the end of the next 10 minute is '\(x\)'. Assuming that Newton's law of cooling is applicable, the value of '\(x\)' will be

• A. \(\frac{9}{5}θ\)
• B. \(\frac{7}{4}θ\)
• C. \(\frac{3}{2}θ\)
• D. \(\frac{4}{3}θ\)

Hint:

Excess temperature over the room decays exponentially. It halves in the first 10 minutes, so it halves again in the next 10 minutes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Newton's law of cooling says the rate of fall of temperature is proportional to the excess temperature over the surroundings. This gives an exponential decay of the excess temperature.

Step 2: Key Formula or Approach:
\[ \theta_{\text{excess}}(t) = \theta_{\text{excess}}(0)\,e^{-kt} \]
Equal time intervals give equal ratios of the excess temperature.

Step 3: Detailed Explanation:
The room temperature is \(\theta\).
At the start the excess is \(3\theta - \theta = 2\theta\). After 10 minutes the body is at \(2\theta\), so the excess is \(2\theta - \theta = \theta\).
The ratio over 10 minutes is \(\dfrac{\theta}{2\theta} = \dfrac12\).
In the next 10 minutes the excess halves again, from \(\theta\) to \(\dfrac\theta2\).
So the body temperature is
\[ x = \theta + \frac\theta2 = \frac{3}{2}\theta \]
Option (A) \(\tfrac95\theta\) and (B) \(\tfrac74\theta\) would result from the approximate average temperature method, which does not apply exactly when the fall is large. Option (D) \(\tfrac43\theta\) gives a ratio of 3 and not 2.

Final Answer:
The temperature after the next 10 minutes is \(\dfrac32\theta\), option (C). \[ \boxed{\frac{3}{2}\theta \text{ (C)}} \]