CAT 2017 Quantitative Ability

Area of the box = area (2 hexagons) + area (6 rectangles).
$=6\frac{\sqrt{3}}{4}{\mathit{a}}^2\times 2+6\times \mathit{a}\mathit{h}$
$=3\sqrt{3}{\mathit{a}}^2+6\mathit{a}\mathit{h}$
In the revised box, the height becomes = 2h
New area = $6\frac{\sqrt{3}}{4}{\mathit{a}}^2\times 2+6\times \mathit{a}\times 2\mathit{h}$
$=3\sqrt{3}{\mathit{a}}^2+12\mathit{a}\mathit{h}$
Area has increased by 50%
⇒ New area = 1.5 x (old area)
$\Rightarrow \left(3\sqrt{3}{\mathit{a}}^2+6\mathit{a}\mathit{h}\right)15=3\sqrt{3}{\mathit{a}}^2+12\mathit{a}\mathit{h}$
$\Rightarrow 3\sqrt{3}{\mathit{a}}^2\times 15+9\mathit{a}\mathit{h}=3\sqrt{3}{\mathit{a}}^2+12\mathit{a}\mathit{h}$
$\Rightarrow 3\sqrt{3}{\mathit{a}}^2\times 05=3\mathit{a}\mathit{h}$
$\Rightarrow \frac{\sqrt{3}}{2}\mathit{a}=\mathit{h}$
$\Rightarrow \frac{\mathit{a}}{\mathit{h}}=\frac{2}{\sqrt{3}}$
$\Rightarrow \mathit{a}:\mathit{h}=2:\sqrt{3}$