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The arch of a bridge is a parabola. Its highest point is the top of the arch, and the span at the…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

The arch of a bridge is a parabola. Its highest point is the top of the arch, and the span at the ground is $24$ metres. The top is $9$ metres above the ground. Take the origin at the top with the arch opening downward, so the equation is $x^2 = -4ay$ . Find $a$ , the distance of the focus below the top, and the height of the arch above the ground at a place $6$ metres from the centre.

A parabolic bridge arch opening downward, with its top at the origin and its ends at (-12, -9) and (12, -9), drawn in blue. The span of 24 metres and the height of 9 metres are marked. A red vertical line at x equals 6 shows the height of the arch above the ground at that place.

Need a reminder? The key ideaShow

The parabola $y^2 = 4ax$ has its focus at $(a, 0)$ and its directrix $x = -a$ . The latus rectum has length $4a$ , and the focal distance of $(x_1, y_1)$ is $x_1 + a$ . The point $(at^2, 2at)$ is on it for every $t$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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