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: Work with the exponential law:
Let $e^{-10k} = r$ be the ratio of the excess temperature after one interval.

Step 2: Find r:
Excess goes from $3\theta - \theta = 2\theta$ to $2\theta - \theta = \theta$ so $r = \frac12$.

Step 3: Apply twice:
After two intervals the excess is $2\theta\,r^2 = 2\theta\cdot\frac14 = \frac\theta2$.

Step 4: Add the room temperature:
$x = \theta + \frac\theta2 = \frac{3\theta}2$.

Final Answer:
Option (C). \[ \boxed{\frac{3}{2}\theta \text{ (C)}} \]