A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center.
The points (-50, 0) and (50, 0) lie on the parabola on the x-axis since the span is 100.
The points (-40, 10) and (40, 10) also lie on the parabola.
We use for our equation of a parabola with vertex (h, k) since the axis if symmetry is vertical. We know our vertex is at (0, k). We need to find k.
Substituting point (50, 0) into this equation, we get:
Substituting point (40, 10) into this equation, we get:
Use the two boxed equations to solve for p.
Subtract the two equations to get:
Now, to find k, we substitute p back into one of our boxed equations.