To determine for which distances [tex]\( x \)[/tex] the arch is more than 200 feet above the ground, we start with the equation modeling the height of the arch:
[tex]\[ y = -0.0063 x^2 + 4 x, \][/tex]
where [tex]\( x \)[/tex] is the distance in feet from the left foot of the arch, and [tex]\( y \)[/tex] is the height in feet above the ground.
We need to find the values of [tex]\( x \)[/tex] such that the height [tex]\( y \)[/tex] is greater than 200 feet. This translates to solving the inequality:
[tex]\[ -0.0063 x^2 + 4 x > 200. \][/tex]
To solve this inequality, we first set it as an equation to find the boundary points where the height is exactly 200 feet:
[tex]\[ -0.0063 x^2 + 4 x = 200. \][/tex]
Let's rearrange this equation to standard quadratic form:
[tex]\[ -0.0063 x^2 + 4 x - 200 = 0. \][/tex]
Next, we solve this quadratic equation for [tex]\( x \)[/tex]. The solutions to this quadratic equation will be the points where the height of the arch is exactly 200 feet. The solutions to this equation are:
[tex]\[ x = 54.7151530595738 \ \text{feet} \][/tex]
[tex]\[ x = 580.205481861061 \ \text{feet} \][/tex]
These values of [tex]\( x \)[/tex] are the points at which the height of the arch is exactly 200 feet. To determine where the arch is more than 200 feet above the ground, we consider the shape of the quadratic function [tex]\( -0.0063 x^2 + 4 x \)[/tex], which is a downward-facing parabola.
The arch's height will be more than 200 feet between these two [tex]\( x \)[/tex]-values. Therefore, the distances [tex]\( x \)[/tex] for which the height of the arch is more than 200 feet above the ground are:
[tex]\[
54.7151530595738 < x < 580.205481861061.
\][/tex]
Hence, the arch is more than 200 feet above the ground for [tex]\( x \)[/tex] in the interval [tex]\((54.7151530595738 \ \text{feet}, 580.205481861061 \ \text{feet})\)[/tex].
