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Problem of the Week
Problem E
Across the Prism

A trapezoidal prism is a prism in which opposite parallel ends are congruent trapezoids.

For the trapezoidal prism shown, the opposite parallel sides of each trapezoid are \(36\) cm and \(16\) cm in length. The non-parallel sides of each trapezoid are \(16\) cm and \(12\) cm in length. The prism is \(40\) cm long. The volume of the prism is \(9984\) cm\(^3\).

A body diagonal of a prism is a line connecting two vertices that are not on the same face. Find the length of the body diagonal \(AB\) (indicated in the diagram), accurate to \(1\) decimal place.

The prism lies on one of its rectangular faces with one of the trapezoidal faces at the front and the other at the back. Point A is on the front trapezoidal face and is located at one of the vertices of the longer of the two parallel sides. Point B is on the back trapezoidal face and is located at the vertex of the shorter of the two parallel sides that is farthest from Point A. Segment AB passes diagonally through the interior of the prism.