# A road of length 2 metre at zero degree Celsius and having expansion coefficient is equals to 3 x + 2 into 10 raise to the power minus 6 where x is the distance from one end of the road in centimetre find the length of road at 20 degree Celsius

Samyak Jain
333 Points
3 years ago
Take a small element dx of the road at a distance x from one end and we need to find change in length dL of this element for temperature change $\dpi{80} \Delta$T = (20 – 0)$\dpi{80} \degree$C = 20 $\dpi{80} \degree$C

But first we need expansion coefficient for this element which is by definition
the fractional change in length per unit change in temperature.
i.e.  $\dpi{100} \alpha$ = dL / dx $\dpi{80} \Delta$T  =  (3x + 2).10 – 6
i.e.  dL = (3x + 2).10 – 6 dx $\dpi{80} \Delta$T , here $\dpi{80} \Delta$T is constant.

Integrating both sides, we get
$\dpi{80} \Delta$L = $\dpi{80} \Delta$T.10 – 6 (3 x2 / 2  + 2 x) with limits of x from 0 to 200 cm.
$\dpi{80} \Delta$L = $\dpi{80} \Delta$T.10 – 6 (3.2002 / 2  + 2 x 200) = $\dpi{80} \Delta$T.10 – 6 (60000 + 400) = 604 x 10 – 4 $\dpi{80} \Delta$T
= 604 x 10 – 4 x 20 = 1.208 cm
But $\dpi{80} \Delta$L = L – 200 cm , where L is the final length of road at 20 $\dpi{80} \degree$C.
So, L – 200 cm = 1.208 cm
L = 201.208 cm.