Reflection Box

Geometry Level 3

A beam of light is emitted within a rectangular box from point \((x,y) = (3,\frac{1}{2})\) at an angle of \(45^{\circ}\) with respect to the horizontal. The coordinates of the four box corners are \((0,0),(4,0),(4,2),\) and \((0,2)\), as shown in the figure.

The beam of light reflects off of the sides of the box such that the horizontal velocity is reversed when the beam strikes a vertical side, and the vertical velocity is reversed when the beam strikes a horizontal side.

Determine the \(x\) coordinate of the location at which the beam first strikes the bottom side \((y=0)\).

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