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

For Rod A, `alpha_(A)=Deltal//lxxDeltaT=0.50//(25xx100)`
`=2xx10^(-5)//.^(@)C`
For Rod B, `alpha_(B)=Deltal//lxxDeltaT=0.40//(40xx100)`
`=1xx10^(-5)//.^(@)C`
let the composite rod C consists of a length `l_(1)` of rod A and a length `l_(2)` of rod B .
Then, `l_(1)+l_(2)=50`
Total expansion is
`Deltal_(1)+Deltal_(2)=0.03`
`Deltal_(1)=l_(1)alpha_(4)DeltaT`
`=l_(1)xx2xx10^(-5)xx50`
`Deltal_(2)=l_(2)alpha_(B)DeltaT`
`=l_(2)xx1xx10^(-5)xx50`
put these values above we get, ltBrgt `l_(1)xx2xx10^(-5)xx50+l_(2)xx1xx10^(-5)xx50=0.03`
Put `l_(1)=50-l_(2)`
We have
`l_(1)=10cm,l_(2)=40cm`