A pan filled with hot food cools from \( 94^{\circ} \mathrm{C} \) to \( 86^{\circ} \mathrm{C} \) in \( 2 \) minutes. When the room
temperature'is \( 20^{\circ} \mathrm{C} \). How long will it cool from \( 74^{\circ} \mathrm{C} \) to \( 66{ }^{\circ} \mathrm{C} \) ?

Given, food cools from \( 94^{\circ} \mathrm{C} \) to \( 86^{\circ} \mathrm{C} \) in \( 2 \) minutes; room temperature \( =20^{\circ} \mathrm{C} \)
Now, \( \frac{d T}{d t}=k\left(\theta-\theta_{0}\right) \)
For \( 1 \) st case, \( \frac{d T}{d t}=\frac{94-86}{2}=\frac{8}{2}=4^{\circ} \)
and \( \left(\theta-\theta_{0}\right)=\frac{(94+86)}{2}-20^{\circ}=70^{\circ} \)
\( \therefore 4=k \) to \( \Rightarrow k=\frac{4}{70} \)
For \( 2 \) nd case, \( \frac{d T}{d t}=\frac{74-66}{t}=\frac{8}{t} \)
and \( \theta-\theta_{0}=\frac{(74+66)}{2}-20=50^{\circ} \)
\( \therefore \frac{8}{t}=\frac{4}{70} \times 50 \)
\( \Rightarrow t=\frac{8 \times 70}{4 \times 50}=2.8 \)
Thus, food cools from \( 74^{\circ} \mathrm{C} \) to \( 66^{\circ} \mathrm{C} \) in \( 2.8 \) minutes