For a system where Newton's law of cooling is applicable,the initial rate of cooling is $R \ ^\circ C/sec$. Find the time when the temperature difference $\Delta T_0$ (initial temperature difference) is reduced to half.

$A$ body cools in a surrounding which is at a constant temperature of $\theta_0$. Assuming that it obeys Newton's law of cooling,its temperature $\theta$ is plotted against time $t$. Tangents are drawn to the curve at the points $A(\theta = \theta_1)$ and $B(\theta = \theta_2)$. These tangents meet the time-axis at angles $\alpha_1$ and $\alpha_2$ as shown in the graph. Then:

$A$ body cools from $80^{\circ} C$ to $50^{\circ} C$ in $5 \text{ min}$. In the next time of $t \text{ min}$, the body continues to cool from $50^{\circ} C$ to $30^{\circ} C$. The total time taken by the body to cool from $80^{\circ} C$ to $30^{\circ} C$ is
[The temperature of the surroundings is $20^{\circ} C$.] (in $\text{ min}$)

$A$ pan filled with hot food cools from $50^{\circ} C$ to $49.9^{\circ} C$ in $5 \text{ sec}$. How long will it take to cool from $40^{\circ} C$ to $39.9^{\circ} C$, if the room temperature is $30^{\circ} C$ (in $\text{ sec}$)?

$A$ block of metal is heated to a temperature much higher than the room temperature and allowed to cool in a room free from air currents. Which of the following curves correctly represents the rate of cooling?

Two spheres of radii $R_1$ and $R_2$ have densities $\rho_1$ and $\rho_2$ and specific heats $S_1$ and $S_2$. If they are heated to the same temperature,what is the ratio of their rates of cooling?