2023 AIME II Problem 14

A cube-shaped container has vertices A, B, C, and D, where \overline{AB} and \overline{CD} are parallel edges of the cube, and \overline{AC} and \overline{BD} are diagonals of faces of the cube, as shown. Vertex A of the cube is set on a horizontal plane \mathcal{P} so that the plane of the rectangle ABDC is perpendicular to \mathcal{P}, vertex B is 2 meters above \mathcal{P}, vertex C is 8 meters above \mathcal{P}, and vertex D is 10 meters above \mathcal{P}. The cube contains water whose surface is parallel to \mathcal{P} at a height of 7 meters above \mathcal{P}. The volume of water is \frac{m}{n} cubic meters, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.