Math CentralQuandaries & Queries


Question from SANDRA, a student:

a bridge is constructed across the river that is 200 feet wide. the arch is parabolic
so that the focus is on the water. A sheep 50 ft wide and 30 ft high passes safely
through the arch
a) find equation of the arch
b) find the highest point of the arch

Hi Sandra.

The distance from the focus to a point on the parabola is the same as the distance from that point to the directrix.

That means that since the distance from the focus to the foot of the bridge is 100 ft (half of 200), then the directrix is 100 ft up as well.

So 100 ft is the distance from the focus to the directrix as well, since the directrix is parallel to the water and above the bridge. But you also know the vertex is halfway between the focus and the directrix, so that is half of 100 ft. So the bridge is 50ft above the water. That answers part (b).

Now think of a graph where the origin is on the water, directly below the center of the bridge. The parabola's vertex is (0, 50) and it passes through the foot of the bridge at (100,0). You can substitute these numbers into y - k = (-1/[4p])(x-h)2 to solve for p and complete part (a). Did you forget to put the "p" in your question? I think you meant to write it there.

Hope this helps,
Stephen La Rocque.

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