A metal rod (A) of `25 cm` length expands by `0.050 cm` when its temperature is raised from `0^@ C` to `100^@ C`. Another rod (B) of a different metal of length `40 cm` expands by `0.040 cm` for the same rise in temperature. A third rod ( C) of `50 cm` length is made up of pieces of rods (A) and (B) placed end to end expands by `0.03 cm` on heating from `0^@ C` to `50^@ C`. Find the lengths of each portion of the composite rod.

Explanation

From the given data for rod A, we have
`DeltaL=alpha_ALDeltaT`
or `alpha_A=(DeltaL)/(LDeltaT)=(0.05)/(25xx100)=2xx10^(-5)//.^@C`
For rod B, we have `DeltaL=alpha_BLDeltaT`
or `alpha_B=(DeltaL)/(LDeltaT)=(0.04)/(40xx100)=10^(-5)//.^@C`
If rod C is made of segments of rod A and B of lengths `l_1` and `l_2`, respectively, then we have at `0^@` C.
`l_1+l_2=50cm`
At `T=50^@C` `l'_1+l'_2=50.03cm`
Thus `alpha_Al_1DeltaT+alpha_Bl_2DeltaT=0.03cm`
or `2xx10^-5xxl_1xx50+10^-5xxl_2xx50=0.03cm`
`2l_1+l_2=(0.03)/(50)xx10^5=60cm`