derive relation between alpha,beta,gamma related to thermal expansion

This is regarding linear , areal, volume expansion of substances.
ΔL = α L ΔT
Area of a square sheet of dimensions L * L:
= A+ΔA = (L+ α L ΔT)² = L² + 2 α L² ΔT   ignoring α² terms (too small).
ΔA = 2 α A ΔT
β = ΔA/A  = 2 α
Volume = V = L³
Changed volume = V+ ΔV = (L + α L ΔT)³
= L³ + 3 α L³ ΔT  +  terms with α² or α³  ignored (too small)
ΔV = 3 α V ΔT
=> gamma = ΔV/V = 3 α

Relation between alpha ad gammaV= L3L= L.[1+ alpha ∆T]v= L.3(1+alpha∆T)3Then, using binomial theorem(1+x)n = 1+nxTherefore, V= L.3(1+3alpha∆T)V= V.(1+3alpha∆T)V=3V.(1+Y∆T)Y=3alphaAlpha= Y/3Between alpha ad betaS= L2L=L.(1+B∆T)V= L.2(1+B∆T)2Again using binomial theorem (1+x)n=1+nxV= l.3(1+2B∆T)V=V.(1+2B∆T)V=V.(1+B∆t)B=2alphaAlpha = B/2Therefore,Alpha/1 =Beta(B)/2=Gamma(Y)/3Alpha:Beta:GammaHence solve...

Mathematically, we can show that alpha, beta and gamma are related as follows: alpha :beta : gamma =1:2:3