2024 GESS PRELIMS P1 Q11

The diagram shows a prism in which its cross-section is a regular hexagon of side \( \left(5 + \sqrt{3} \right)\) cm.

1. Find an expression for the cross-sectional area of the prism in the form \(q\sqrt{3} + p\) cm\(^2\), where \(p\) and \(q\) are integers.[3]
2. Given that the volume of the prism is \(3 \left(32\sqrt{3} + 138 \right)\) cm\(^3\), find the height of the prism in the form \( \left(a\sqrt{3} + b \right)\) cm, where \(a\) and \(b\) are integers.[3]