The temperature inside the copper rod varies linearly with the distance from the hot end of the rod. This means that we can find the temperature at 23 cm (let's call it 'point A') from the cool end by solving a linear proportion.
The temperature difference between the two ends of the rod is
[tex]\Delta T = 105^{\circ}-21^{\circ} = 84^{\circ}[/tex]
and this corresponds to a length of 81 cm. Therefore, we can write:
[tex]84^{\circ}:81~cm = x:23~cm[/tex]
from which we find
[tex]x=23.8~^{\circ}[/tex]
This is not the final answer actually; this is the temperature difference between the cool end and point A. So, the temperature at point A is
[tex]T_A = 21^{\circ}+x=44.8^{\circ}[/tex]
