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(A) Let the thermal resistance of each rod be $R$ and the temperature of the junction be $T_X$.According to the principle of conservation of energy,th

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$60$

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(A) Let the thermal resistance of each rod be $R$ and the temperature of the junction be $T_X$.According to the principle of conservation of energy,the total heat flowing into the junction must equal the total heat flowing out of the junction.Here,heat flows from the two rods at $90^{\circ} C$ towards the junction $X$,and then from the junction $X$ towards the rod at $0^{\circ} C$.Let $H_1$ and $H_2$ be the heat currents from the two rods at $90^{\circ} C$,and $H$ be the heat current flowing towards the $0^{\circ} C$ end.$H = H_1 + H_2$Using the formula for heat current $H = \frac{\Delta T}{R}$,we have:$\frac{T_X - 0}{R} = \frac{90 - T_X}{R} + \frac{90 - T_X}{R}$Since the rods are identical,the thermal resistance $R$ is the same for all.$T_X = (90 - T_X) + (90 - T_X)$$T_X = 180 - 2T_X$$3T_X = 180$$T_X = 60^{\circ} C$Thus,the temperature at the junction $X$ is $60^{\circ} C$.

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