A semicircular rod is connected at its ends to a straight rod made...

Let the radius of the semicircular rod be $R$.

The length of the straight rod is equal to the diameter of the semicircle, so $L_1 = 2R$.

The length of the semicircular rod is $L_2 = \pi R$.

Let the temperatures at the two junctions be $T_1$ and $T_2$. The rate of heat transfer through a rod of length $L$, cross-sectional area $A$, and thermal conductivity $k$ is given by $H = \dfrac{kA(T_1 - T_2)}{L}$.

The heat transferred through the semicircular rod in a given time interval $t$ is:

$Q_2 = \dfrac{kA(T_1 - T_2)t}{\pi R}$

The heat transferred through the straight rod in the same time interval $t$ is:

$Q_1 = \dfrac{kA(T_1 - T_2)t}{2R}$

The ratio of the heat transferred through the semicircular rod to that through the straight rod is:

$\dfrac{Q_2}{Q_1} = \dfrac{\dfrac{kA(T_1 - T_2)t}{\pi R}}{\dfrac{kA(T_1 - T_2)t}{2R}} = \dfrac{2R}{\pi R} = \dfrac{2}{\pi}$

The required ratio is $2 : \pi$.