Question

A bridge is to be built in the shape of a parabola and is to have a span of 140 feet. The height of the arch at a distance of 50 feet from the center is to be 20 feet. Find the height of the arch at its center.

Answer 1

Graphically the parabolic bridge can be represented by an inverted parabola symmetrically positioned so that its axis of symmetry is the y-axis with vertex at (0,h) where h feet is the height of the bridge at its centre.

The points (50,20) and (-50,20) lie on the parabola because of symmetry about the y-axis, and the span of the bridge is represented by (70,0) and (-70,0), the x-intercepts. The difference between the intercepts is 140 feet, the span of the bridge.

If the equation of the parabola is given by the general formula:

y=-ax²+h (which has vertex (0,h)) then we know that the x-intercepts can be plugged in:

0=-4900a+h so h=4900a. (a is negative because the parabola is inverted.)

We can also plug in the given points:

20=-2500a+4900a=2400a, a=20/2400=1/120, h=4900/120=245/6 (the central height).

Therefore y=245/6-x²/120 is the equation of the parabola, with vertex at (0,245/6), so the height at the centre is 245/6 feet or 40ft 10in.