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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

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
 

Grade:12th pass

1 Answers

Samyak Jain
333 Points
2 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 \DeltaT = (20 – 0)\degreeC = 20 \degreeC
 
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.  \alpha = dL / dx \DeltaT  =  (3x + 2).10 – 6
i.e.  dL = (3x + 2).10 – 6 dx \DeltaT , here \DeltaT is constant.
 
Integrating both sides, we get
\DeltaL = \DeltaT.10 – 6 (3 x2 / 2  + 2 x) with limits of x from 0 to 200 cm.
\DeltaL = \DeltaT.10 – 6 (3.2002 / 2  + 2 x 200) = \DeltaT.10 – 6 (60000 + 400) = 604 x 10 – 4 \DeltaT
      = 604 x 10 – 4 x 20 = 1.208 cm
But \DeltaL = L – 200 cm , where L is the final length of road at 20 \degreeC.
So, L – 200 cm = 1.208 cm
L = 201.208 cm.

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