A hot body placed in air is cooled down according to Newtons Law of cooling, the rate of decrease of temperature being  $\beta$ times the temperature difference from the surrounding. Starting from $t=0$ , the time in which the rate of cooling is half of its maximum value

According to Newton's law of cooling, the rate of cooling is directly proportional to the temperature difference of the body to the surroundings.

  $\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{\beta }(\mathrm{T}-{\mathrm{T}}_0)$

Initially the rate of cooling will be maximum, the rate of cooling will be halved when the temperature difference between the body and surroundings is halved.

  $\frac{1}{\beta }\int \frac{dT}{T-T_0}=-\int dt$

${\left[\frac{1}{\mathrm{\beta }}\ln \left(\mathrm{T}-{\mathrm{T}}_0\right)\right]}_{{\mathrm{T}}_1}^{{\mathrm{T}}_2}=-\mathrm{t}$

${\left[\frac{1}{\mathrm{\beta }}\ln \frac{\left({\mathrm{T}}_2-{\mathrm{T}}_0\right)}{\left({\mathrm{T}}_1-{\mathrm{T}}_0\right)}\right]}_{{\mathrm{T}}_1}^{{\mathrm{T}}_2}=-\mathrm{t}$

Here, $\frac{\left({\mathrm{T}}_2-{\mathrm{T}}_0\right)}{\left({\mathrm{T}}_1-{\mathrm{T}}_0\right)}=\frac{1}{2}$

So, the time in which the rate of cooling is half of the maximum rate is given by $\mathrm{t}=\frac{\ln \left(2\right)}{\mathrm{\beta }}$