Find Maximum Volume Of Ruled Frustum

Geometry Level 5

The solid figure shown has a height of 2, with a square top of area 2 and a circle base of area \(\pi\), the centers of which lie on the same axis that both faces are perpendicular to. The surface that extends from the top face to bottom face is a ruled surface, which is a surface where any point on it lies on a line of points also on the same surface extending from the edge of the top face to the edge of the bottom face. No point on the surface lies on more than one such line.

The maximum volume of this solid can be expressed as

\[ \dfrac{1}{a}\left(b+c\sqrt{d}+e\pi\right), \]

where \(a, b, c, d\) and \(e\) are all positive integers, \(d\) is square-free, and \(a\) and \(b\) are coprime. Find the sum \(a+b+c+d+e\).

Details and Assumptions:

  • There can be more than 1 kind of ruled surface that spans from the top to bottom face.

  • A finite few points may lie more than one line on the ruled surface, and by "maximum", it may or may not only be a supremum.


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