Question

A metal rod cools at the rate of $4^\circ\text{C}/\text{min}$ when its temperature is $90^\circ\text{C}$ and the rate of $1^\circ\text{C}/\text{min}$ when its temperature is $30^\circ\text{C}$. The temperature of the surrounding is

• A. 20^\circ\text{C}
• B. 15^\circ\text{C}
• C. 10^\circ\text{C}
• D. 5^\circ\text{C}

Hint:

The rate of cooling is directly proportional to the temperature excess $(\theta - \theta_0)$.

Answer 1

The Correct Option is C

Step 1: Apply Newton's Law of Cooling, which states that the rate of cooling $R$ is proportional to the difference in temperature between the body $\theta$ and its surroundings $\theta_0$. \[ R = k(\theta - \theta_0) \]
Step 2: Set up the ratio for the two different cooling conditions provided. \[ \frac{R_1}{R_2} = \frac{\theta_1 - \theta_0}{\theta_2 - \theta_0} \]
Step 3: Substitute the given values: $R_1 = 4$, $\theta_1 = 90$, $R_2 = 1$, and $\theta_2 = 30$. \[ \frac{4}{1} = \frac{90 - \theta_0}{30 - \theta_0} \]
Step 4: Cross-multiply to solve the equation for $\theta_0$. \[ 4(30 - \theta_0) = 90 - \theta_0 \] \[ 120 - 4\theta_0 = 90 - \theta_0 \]
Step 5: Rearrange the terms to find the value of $\theta_0$. \[ 120 - 90 = 4\theta_0 - \theta_0 \] \[ 30 = 3\theta_0 \] \[ \theta_0 = 10^\circ\text{C} \]