parabolic arch

Question

A parkway 80ft. wide is spanned by a parabolic arch 100ft. long along the horizontal. If the parkway is in the center, how must high the vertex of the arch be in order to give a minimum clearance of 20ft. over the parkway?

Answer 1

Let y=-ax²+H, where this centrally located inverted parabola represents the arch with vertex at (0,H), and H is the height of the vertex above the parkway.

The y value represents the horizontal positions of the walls of the arch from the centre of the parkway and therefore from the vertex. When x=50ft or -50ft, y=0, so 2500a=H; when x=40ft or -40ft, y=20ft (the minimum clearance or headroom), 20=-1600a+H=-1600a+2500a=900a, so a=20/900=1/45.

y=-2x²/45+H. H=2500a=2500/45=500/9=55.56ft or 55ft 6⅔in. The arch is represented below in red. The green lines indicate the height of the vertex and the minimum height of 20ft.