According to Newton's law of cooling,the rate of cooling of a body is proportional to $(\Delta \theta )^n$,where $\Delta \theta$ is the difference of the temperature of the body and the surroundings,and $n$ is equal to

If a body cools from a temperature of $62^{\circ} C$ to $50^{\circ} C$ in $10 \, \text{minutes}$ and to $42^{\circ} C$ in the next $10 \, \text{minutes}$, then the temperature of the surroundings is (in $^{\circ} C$)

$A$ can is taken out from a refrigerator at $0^{\circ}C$. The atmospheric temperature is $25^{\circ}C$. If $t_1$ is the time taken to heat from $0^{\circ}C$ to $5^{\circ}C$ and $t_2$ is the time taken to heat from $10^{\circ}C$ to $15^{\circ}C$,then:

One day in the morning,Ramesh filled up $\frac{1}{3}$ of a bucket with hot water from a geyser to take a bath. The remaining $\frac{2}{3}$ was to be filled with cold water (at room temperature) to bring the mixture to a comfortable temperature. Suddenly,Ramesh had to attend to some work that would take $5-10 \text{ min}$ before he could take a bath. He had two options: $(1)$ fill the remaining bucket completely with cold water and then attend to the work,$(2)$ first attend to the work and then fill the remaining bucket just before taking a bath. Which option do you think would have kept the water warmer? Explain.

$A$ 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 $\text{ sec}$)?

$A$ body cools from a temperature $3\theta$ to $2\theta$ in $10 \text{ minutes}$. The room temperature is $\theta$. The temperature of the body at the end of the next $10 \text{ minutes}$ is '$x$'. Assuming that Newton's law of cooling is applicable, the value of '$x$' will be