A bucket full of hot water is kept in a room. If it cools from \(75^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) in \(\mathrm{t}_1\) minutes, from \(70^{\circ} \mathrm{C}\) to \(65^{\circ} \mathrm{C}\) in \(\mathrm{t}_2\) minutes and \(65^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) in \(\mathrm{t}_3\) minutes, then

According to Newton's law of cooling, Rate of cooling \(\propto\) Mean temperature difference \(\Rightarrow \frac{\text { Fall in temperature }}{\text { Time }(\mathrm{t})} \propto\left(\frac{\theta_1+\theta_2}{2}-\theta_0\right)\)
Case 1: \(\left(\frac{\theta_1+\theta_2}{2}\right)_1=\left(\frac{75+70}{2}\right)_1=72.5\)
Case 2: \(\left(\frac{\theta_1+\theta_2}{2}\right)_2=\left(\frac{70+65}{2}\right)_2=67.5\)
Case 3: \(\left(\frac{\theta_1+\theta_2}{2}\right)_3=\left(\frac{65+60}{2}\right)_3=62.5\)
\(\begin{aligned}
\therefore \quad & \left(\frac{\theta_1+\theta_2}{2}\right)_1\gt\left(\frac{\theta_1+\theta_2}{2}\right)_2\gt\left(\frac{\theta_1+\theta_2}{2}\right)_3 \\
& \Rightarrow \mathrm{t}_1 \lt \mathrm{t}_2 \lt \mathrm{t}_3
\end{aligned}\)