Answer:
x = 12
Side length = [tex]2\sqrt{3}[/tex] in
Step-by-step explanation:
Area of the given hexagon = 6 × (Area of the triangular section)
Area of the triangular section = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]
= [tex]\frac{1}{2}(2)(\sqrt{3})^{\frac{x}{12}}(\sqrt{3})^{\frac{x}{6}}[/tex]
= [tex](\sqrt{3})^{\frac{x}{12}}[(\sqrt{3})^2]^{\frac{x}{12}}[/tex]
= [tex](\sqrt{3})^{\frac{x}{12}}(3)^{\frac{x}{12}}[/tex]
= [tex](3\sqrt{3})^{\frac{x}{12}}[/tex]
Now area of the given hexagon = [tex]6(3\sqrt{3})^{\frac{x}{12}}[/tex]
Since, area of the hexagon is = [tex]18\sqrt{3}[/tex] in²
[tex]6(3\sqrt{3})^{\frac{x}{12}}=18\sqrt{3}[/tex]
[tex](3\sqrt{3})^{\frac{x}{12}}=(3\sqrt{3})^1[/tex]
[tex]\frac{x}{12}=1[/tex]
x = 12
Therefore, side length = [tex]2(\sqrt{3})^{\frac{x}{12}}[/tex]
= [tex](2\sqrt{3})^{\frac{12}{12}}[/tex]
= [tex]2\sqrt{3}[/tex] in.