Stage Truss Projection Screen Aspect Ratio (Entertainment Engineering & Design)

In entertainment engineering and theatrical production design for large-scale Las Vegas Strip residency productions, structural rigging engineers design stage proscenium arches using modular aluminum box trusses. Suspended high-resolution LED video tile displays are inscribed within these truss boundaries to create immersive visual effects without obstructing overhead theatrical lighting instruments, pyrotechnic rigs, or line arrays.

An entertainment engineering team at UNLV is staging an arena concert. The main performance stage features an overhead parabolic structural truss arch anchored to the stage floor ($y = 0$) with its centerline along the $y$-axis. The internal clearance boundary of the truss is modeled by the quadratic parabola:

                            y (meters)
                                |
                             9 -|- Apex
                               /|\
                              / | \   Parabolic Lighting Truss Arch
         +-------------------/--+--\-------------------+
         | Screen Corner    /|  |  |\   Screen Corner  |
         | (-x, y)         / |  |  | \  (x, y)         |
         |                /  |  |  |  \                |
         |               /   |  |  |   \               |
         |              /    |  |  |    \              |
   ======+=============+=====+==+==+=====+=============+====== x (meters)
       Stage Floor   -x      0     x

A massive rectangular LED video screen must be centered horizontally on the stage, with its lower edge resting on the stage floor ($y = 0$) and its upper two corners touching the inner chord of the parabolic truss at $(-x, y)$ and $(x, y)$.

1. Express the total screen width, total screen height, and screen area $A$ in terms of the horizontal coordinate $x$ of the upper right corner.
2. Determine the physical Context-Driven Domain for the half-width $x$ bounded by the truss ground footings.
3. Formulate the single-variable objective function $A(x)$ and locate all interior critical points.
4. Execute the Closed Interval Method algorithm (guaranteed by the Extreme Value Theorem) to determine the dimensions (width and height) that maximize the video display area.
5. Calculate the maximum screen area and state the resulting aspect ratio (width-to-height) of the optimized video display.