From: Mike, a secondary school student

A bridge over a river is supported by a parabolic arch...arch is 200 m wide at water level...the maximum height of the arch is 80 m..what is the height of the arch measured from a point on the water 40m from the center of the arch?


Hi Mike.

I've drawn a parabolic arch:

You know that the arch is 80m high. If we choose to draw this on a graph or represent it in an equation, we may as well call the peak of the arch the vertex and put it on the y axis at position (0, 80). That means the ground is along the x = 0 line (the x axis). The distance between the feet of the arch is 200 m, so that means that the feet are at positions (+/-100, 0).

Knowing this, you can create an equation for the parabola. Remember the standard form of a parabola is y = a(x - h)2 + k. where (h, k) represents the vertex of the parabola and a controls the shape and vertical direction.

You need to solve for a by substituting in one of the points (+/-100, 0) in for x and y. This gives you the equation of the parabola.

Since the center of the arch is on the y axis, 40 m from the center of the arch is just x = +/-40 (you can choose either since the parabola is symmetric). So plug 40 in for x and get y, the height of the parabolic arch at this point.

Hope this helps,
Stephen La Rocque.