Question

A body cools from a temperature $3T$ to $2T$ in $10\, minutes$. The room temperature is $T$. Assume that Newton�s law of cooling is applicable. The temperature of the body at the end of next $10\,minutes$ will be -

• A. $\frac{7}{4} T$
• B. $\frac{3}{2} T$
• C. $\frac{4}{3} T$
• D. $T$

Answer 1

The Correct Option is B

To solve this problem, we use Newton's law of cooling, which describes the rate of heat loss of a body to its surroundings.

According to Newton's law of cooling, the rate of change of temperature of the body is proportional to the difference between the temperature of the body and the ambient temperature (room temperature). Mathematically, it can be expressed as:

\[\frac{d\theta}{dt} = -k(\theta - T_r)\]

where:

• \(\theta\) is the temperature of the body at time \(t\).
• \(T_r\) is the room temperature.
• \(k\) is a positive constant.

Given:

• Initial temperature of the body: \(3T\)
• Temperature after 10 minutes: \(2T\)
• Room temperature: \(T\)

First, we calculate the cooling constant \(k\).

\[\frac{2T - 3T}{10} = -k\left(\frac{2T + 3T}{2} - T\right)\] \[-\frac{T}{10} = -k\left(\frac{5T}{2} - T\right)\] \[-\frac{T}{10} = -k\left(\frac{3T}{2}\right)\] \[k = \frac{1}{15}\]

Next, find the temperature after the next 10 minutes.

After 10 more minutes, let the temperature be \(\theta_2\). Applying the same law:

\[\frac{\theta_2 - 2T}{10} = -\frac{1}{15}\left(\frac{\theta_2 + 2T}{2} - T\right)\] \[-\frac{1}{10}(\theta_2 - 2T) = -\frac{1}{15}\left(\frac{\theta_2 + 2T}{2} - T\right)\] \[\theta_2 - 2T = \frac{2}{3}\left(\frac{\theta_2}{2} + T - T\right)\] \[\theta_2 - 2T = \frac{\theta_2}{3}\] \[\theta_2 = \frac{3}{2}T\]

Therefore, the temperature of the body at the end of the next 10 minutes will be \(\frac{3}{2}T\).

Thus, the correct answer is: \(\frac{3}{2}T\).