$\frac{{{\theta _1} - {\theta _2}}}{t} =  - K\left[ {\frac{{{\theta _1} + {\theta _2}}}{2} - {\theta _0}} \right]$

where ${\theta _0}$ is the temperature of surrounding. Now, hot water cools from ${60^ \circ }C\,to\,{50^ \circ }C$ in $10\,minutes,$

$\frac{{60 - 50}}{{10}} =  - K\left[ {\frac{{60 + 50}}{2}{\theta _0}} \right]\,\,\,\,\,\,\,\,\,\,...\left( i \right)$

Agian, it cools from ${50^ \circ }C\,to\,{42^ \circ }C$ in next $10\,minutes.$

$\frac{{50 - 42}}{{10}} =  - K\left[ {\frac{{50 + 42}}{2} - {\theta _0}} \right]\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$

Dividing equations $(i)$ by $(ii)$ we get

$\frac{1}{{0.8}} = \frac{{55 - {\theta _0}}}{{46 - {\theta _0}}}$

$\frac{{10}}{8} = \frac{{55 - {\theta _0}}}{{46 - {\theta _0}}}$

$460 - 10{\theta _0} = 440 - 8{\theta _0}$

$2{\theta _0} = 20$

${\theta _0} = {10^ \circ }c$