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A metal rod cools at the rate of $4{ }^{\circ} \mathrm{C} / \mathrm{min}$ when its temperature is $90^{\circ} \mathrm{C}$ and the rate of $1{ }^{\circ} \mathrm{C} / \mathrm{min}$ when its temperature is $30^{\circ} \mathrm{C}$. The temperature of the surrounding is
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The correct answer is:
$10^{\circ} \mathrm{C}$
According to Newton's law of cooling,
$\begin{array}{ll}
& \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{\theta_1-\theta_0}{\theta_2-\theta_0} \\
\therefore \quad & \frac{4}{1}=\frac{90-\theta_0}{30-\theta_0} \\
\therefore \quad & 120-90=4 \theta_0-\theta_0 \\
\therefore \quad & 3 \theta_0=30
\end{array}$
$\therefore \quad$ Temperature of the surroundings is,
$\theta_0=10^{\circ} \mathrm{C}$
$\begin{array}{ll}
& \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{\theta_1-\theta_0}{\theta_2-\theta_0} \\
\therefore \quad & \frac{4}{1}=\frac{90-\theta_0}{30-\theta_0} \\
\therefore \quad & 120-90=4 \theta_0-\theta_0 \\
\therefore \quad & 3 \theta_0=30
\end{array}$
$\therefore \quad$ Temperature of the surroundings is,
$\theta_0=10^{\circ} \mathrm{C}$
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