A glass window is to be fit in an aluminium frame. The temperature in the working day is ${40}^0\mathrm{C}$ and the glass window measures exactly 20cm x 30cm.

$\left[{\mathrm{\alpha }}_{\mathrm{glass}}=9\times {10}^{-6}/{}^0\mathrm{C}, {\mathrm{\alpha }}_{\mathrm{aluminum}}=24\times {10}^{-6}/{}^0\mathrm{C}\right]$ The size of the aluminium frame is____, so that there is no stress on the glass in winter even if the temperature drops to $0^0\mathrm{C}$

Glass                                                            Aluminium

$$\begin{gathered} \mathrm{At} {40}^0\mathrm{C} {\mathrm{L}}_1=20 \mathrm{cm} {\mathrm{L}}_2=? \\ {\mathrm{B}}_1=30 \mathrm{cm} {\mathrm{B}}_2=? \\ \mathrm{At} 0^0\mathrm{C} {\mathrm{L}}_1^1= {\mathrm{L}}_2^1= \\ {\mathrm{B}}_1^1= {\mathrm{B}}_2^1= \\ \mathrm{No} \mathrm{Stress}\Rightarrow {\mathrm{L}}_1^1={\mathrm{L}}_2^1 \& {\mathrm{B}}_1^1={\mathrm{B}}_2^1 ...\left(1\right) \end{gathered}$$

$$\begin{gathered} {\mathrm{L}}_1^1={\mathrm{L}}_1\left[1+{\mathrm{\alpha }}_1\left(-40\right)\right] ....\left(2\mathrm{a}\right) \\ {\mathrm{L}}_2^1={\mathrm{L}}_2\left[1+{\mathrm{\alpha }}_2\left(-40\right)\right].......\left(2\mathrm{b}\right) \\ {\mathrm{B}}_1^1={\mathrm{B}}_1\left[1={\mathrm{\alpha }}_1\left(-40\right)\right].....\left(2\mathrm{c}\right) \\ {\mathrm{B}}_2^1={\mathrm{B}}_2\left[1={\mathrm{\alpha }}_2\left(-40\right)\right]....\left(3\mathrm{d}\right) \\ \mathrm{From} \left(1\right) \& \left(2\right) \mathrm{equation} \\ {\mathrm{L}}_2=20.012 \mathrm{cm}; {\mathrm{B}}_2=30.01 \mathrm{cm} \end{gathered}$$