3.0 Relation between $\alpha ,\,\beta \,$ and $\gamma $

Consider an isotropic solid, cube of length $L$ . When the temperature raises by $\Delta T$ the length increases by $\Delta L$ and area increases by $\Delta A$, then$$\begin{equation} \begin{aligned} dA = \left( {\frac{{dA}}{{dl}}} \right).dl = 2l.dl \\ dl = l\alpha \Delta T \\ dA = 2l.l\alpha \Delta T = \left( {2\alpha } \right)AdT \\ and \\ dA = \beta AdT \\ \Rightarrow \beta = 2\alpha \\\end{aligned} \end{equation} $$ Similarly, $$\gamma = 3\alpha $$Therefore, $$\alpha :\beta :\gamma = 1:2:3$$