Because the rods are made of same material, material properties such as specific heat capacity (#c#), mass density (#\rho#) and coefficient of volume expansion (#\gamma#) are the same.
If #Q# is the heat entering a body of mass #m#, volume #V#, density #\rho# and specific heat capacity #c#, then the change in temperature of the body #\Delta T# is related to other quantities as -
#Q = mc\Delta T = (\rho.V)c\Delta T; \qquad#
Rearranging,
#V\DeltaT = (\frac{Q}{\rho.c})# ...... (1)
Since #\rho# and #c# are material constants, the the product of volume and change in temperature (#V\DeltaT#) will be a constant, if #Q# is a constant.
For a temperature change of #\Delta T# the volume expansion is -
#\Delta V = \gamma (V\DeltaT)# ...... (2)
Let #V_1# and #V_2# be the volumes of the two rods and #\Delta T_1# and #\Delta T_2# are the change in their temperatures when they absorb the same heat #Q#. Then the change in their volumes are -
#\DeltaV_1 = \gamma(V_1\DeltaT_1); \qquad \DeltaV_2 = \gamma (V_2\DeltaT_2)#
But we know from (1): #\quad V_1\DeltaT_1 = V_2\DeltaT_2#
Therefore, #\quad \DeltaV_1 = \DeltaV_2; \qquad \rightarrow \DeltaV_1 : \DeltaV_2 = 1:1#.
