MHTCET 10 May 2024 Shift 2 Paper

Concept:Newton's law of cooling states that the rate of cooling of a body is proportional to the difference between its mean temperature and the surrounding temperature.Explanation:For a small time interval, Newton's law of cooling can be written as:$$\frac{T_1 - T_2}{t} = K \left( \frac{T_1 + T_2}{2} - T_0 \right)$$Here, $T_0 = \theta$ is the room temperature, $K$ is the cooling constant, and $t$ is the time.First, the body cools from $4\theta$ to $3\theta$ in $t$ minutes.Substitute these values:$$\frac{4\theta - 3\theta}{t} = K \left( \frac{4\theta + 3\theta}{2} - \theta \right)$$Simplifying,$$\frac{\theta}{t} = K \cdot \frac{5\theta}{2}$$Therefore, the cooling constant is:$$K = \frac{2}{5t}$$Let the temperature after the next $t$ minutes be $x$.Now apply the same law for the cooling from $3\theta$ to $x$:$$\frac{3\theta - x}{t} = \frac{2}{5t} \left( \frac{3\theta + x}{2} - \theta \right)$$Cancelling $t$ from both sides,$$3\theta - x = \frac{x + \theta}{5}$$Multiply by 5:$$15\theta - 5x = x + \theta$$Rearrange:$$14\theta = 6x$$Hence,$$x = \frac{14\theta}{6} = \frac{7\theta}{3}$$Answer:The temperature of the body after the next $t$ minutes is $\frac{7\theta}{3}$.Therefore, the correct option is D.