If y is our height in the function, in order to find where the x values where y = 200, we sub in accordingly: [tex] 200=-.0063x^2+4x [/tex]. In order to solve for these x values, we now have to factor this. Start by moving the 200 over to the other side by subtraction. Using the quadratic formula is the only way to do this easily. [tex] x=\frac{-4+/-\sqrt{4^2-4(-.0063)(-200)}}{2(-.0063)} [/tex]. Now we simplify a bit:
[tex] x=\frac{-4+/-\sqrt{16-5.04}}{-.0126} [/tex]
and a bit more
[tex] x=\frac{-4+/-\sqrt{10.96}}{-.0126} [/tex]. We then have 2 solutions that are found in
[tex] x=\frac{-4+3.3105}{-.0126} [/tex] and [tex] x=\frac{-4-3.3105}{-.0126} [/tex]. Solving the first one gives us an x value of 57.72 feet, and solving the second one gives us an x value of 580.20 feet. Those are the x values where the arch is 200 feet above the ground. It will be on either side equally spaced from the vertex, which will be the highest point.
