Show that the coefficient of area expansions, `(DeltaA//A)//DeltaT`, of a rectangular sheet of the solid is twice its linear expansively, `alpha_(l).(alpha_(l)=10^(-5)K^(-1))`

Explanation

Consider a rectangular sheet of the solid material of length a and breadth b (Fig. 11.8). When the temperature increases by `Delta T, a ` Increases by `Delta a = alpha_(1) a Delta T` and b increases by `Delta b = alpha_(1), b Detla T`. From Fig. 11.8, the increase in area
`Delta A = DeltaA_(1) + DeltaA_(2) + DeltaA_(3)`
`DeltaA = a Deltab + b Delta a + (Delta a) (Deltab)`
` = a alpha_(1) b DeltaT + b alpha_(1) a DeltaT + (alpha_(1))^(2) ab (Delta T)^(2)`
` = alpha_(1) ab DeltaT (2 +alpha_(1) DeltaT) = alpha_(1) A DeltaT (2 + alpha_(1) DeltaT)`
Since `alpha_(1) approx 10^(-5) K^(-1)`, from Table 11.1 , the product `alpha_(1) Delta T ` for fractional temperature is small in comparision with 2 and may be neglected.
Hence,
`((Delta A)/A) 1/(Delta T) approx 2alpha_(l) `