Show that the coefficeint of area expansions `(Delta A//A)//Delta T` of a rantangular sheet of the solid is twice its linear expansitivity `alpha`.

Explanation

Consider a rectangular sheet of a solid material of length a and breadth b as shown in fig. Area of sheet A = ab . When the temperature of sheet is raised by `Delta T`, let increase in length of sheet be `Delta a` and increase in breadth of sheet be `Delta b`. If `alpha` is the coefficient of linear expansion, then

`Delta a = alpha a Delta T` and `Delta b = alpha b Delta T`
The increase in the area of sheet is
`Delta A = (a + Delta a) (b+Delta b) - ab`
`= aDelta b + b Delta a + Delta a Delta b`
` = a alpha b DeltaT + b alpha a Delta T +alpha^(2) ab (Deltat)^(2)`
` = alpha ab Delta T ( 2+ alpha Delta T)`
` = alpha A Delta T ( 2+ alpha Delta T)`
Since, `alpha` is a small quantity, hence the product `alpha Delta T` for small change in temperatrue is very very small, can be neglected in comparisonto 2.
`:. Delta A = alpha A Delta T 2`
or `((Delta A)/(A Delta T)) = 2 alpha` (proved).