Question

Semi elliptical Arch Bridge: A bridge is built in the shape of a semi elliptical arch. The bridge has a span of $120$feet and a

maximum height of $25$feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances

of $10,30$and $50$feet from the center.

Short answer

The height of the arch at distances $10,30$and $50feet$ from the center are$24.65,21.65$and$13.82feet$.

Step-by-step solution

Step 1. Given information.

The span of the bridge is given as $120feet$.So, the major axis

$$\begin{gathered} 2a=120 \\ a=60 \end{gathered}$$

The maximum height of the bridge is $25$feet. So, semi minor axis $b=25$

The height of the arch at the distance of $10feet$from the center.

. Height of the arch at x=10feet

Substitute $a=60$,$b=25$and $x=10$we get

role="math" localid="1646811943374" $$\begin{gathered} \frac{{10}^2}{{60}^2}+\frac{y^2}{{25}^2}=1 \\ \frac{100}{3600}+\frac{y^2}{625}=1 \\ \frac{y^2}{625}=1-\frac{100}{3600} \\ \frac{y^2}{625}=0.9722 \\ {y}^2=607.625 \\ y=\sqrt{607.625} \\ y=24.65 \end{gathered}$$

Step 3. Height of the arch at x=30 feet

$$\begin{gathered} \frac{{30}^2}{{60}^2}+\frac{y^2}{{25}^2}=1 \\ \frac{900}{3600}+\frac{y^2}{625}=1 \\ \frac{y^2}{625}=1-\frac{900}{3600} \\ \frac{y^2}{625}=0.75 \\ {y}^2=468.75 \\ y=21.65feet \end{gathered}$$

Step 4.  Height of the arch at x=50 feet

$$\begin{gathered} \frac{{50}^2}{{60}^2}+\frac{y^2}{{25}^2}=1 \\ \frac{2500}{3600}+\frac{y^2}{625}=1 \\ \frac{y^2}{625}=1-\frac{2500}{3600} \\ \frac{y^2}{625}=0.3056 \\ {y}^2=191 \\ y=\sqrt{191} \\ y=13.82feet \end{gathered}$$

The height of the arch at distance $10,30$and $50$feet from the center are$24.65,21.65$and$13.82$feet.