one question

consider a square plate of a material whose coeficent of linear expan.is α
and that of superficial exp. is β.let the increase in temp be Δt
let original each side of square be a and the area be A

new length after increase in length = a(1+αΔt)

new area after increase in length = a²(1+αΔt)²

= a²(1+2αΔt)..(1)

new area = A(1+βΔt)...(2)

now A = a²

comparing (1) and (2) we get

β = 2α

its okay if you are desperate but theres no point in starting two threads for same question

if its your doubt
did you try to get the result by differentiation?