2784.11 – Big Stone Arch

The arch of a stone bridge has the form of a semi-ellipse. The straight horizontal span is forty meters, and the maximum height between the span and the arch is ten meters. Find the height, in meters, of the arch ten meters from one end of the span.

5 meters
5 $\surd$ 3 meters
2.5 meters
7 meters
8.667 meters

Solution

Referring to the figure below, we are trying to find the height of the bridge at the point where $x = 10$ , assuming that the center of the span lies at the origin. The equation of the whole ellipse is:

$\frac{x^2}{20^2} + \frac{y^2}{10^2} =1.$

When $x=10$ , then

\begin{aligned} \frac{10^2}{20^2} + \frac{y^2}{10^2} =1 &\leadsto& \frac{1}{4}+\frac{y^2}{10^2} =1 \\ &\leadsto& \frac{y^2}{10^2} =\frac{3}{4} \\ &\leadsto& y^2=\frac{300}{4}=75 \\ &\leadsto& y=\sqrt{75} \approx 8.667. \end{aligned}

The correct answer is e.

It is not clear whether the ellipse itself carries the weight of the road on top of it--the elliptical arch might be merely ornamental. But if you do Google Images for elliptical arch you will see plenty of bridges that seem to have elliptical arches with keystones and that seem to be plenty strong.