Question

Semi elliptical Arch Bridge: An arch for a bridge over a

highway is in the form of half an ellipse. The top of the arch is

$20$feet above the ground level (the major axis). The highway

has four lanes, each $12$feet wide; a center safety strip $8$feet

wide; and two side strips, each $4$feet wide. What should the

span of the bridge be (the length of its major axis) if the

height $28$feet from the center is to be $13$feet?

Short answer

The span of the bridge is$73.69feet$.

Step-by-step solution

Step. 1 Given information. 

The maximum height of the arch at the center is $20feet$and the height at $28feet$from the center is $13feet$.

The equation of ellipse is

localid="1647076670481" $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Step 2. To find span of the bridge.

Here, $b=20$and the ellipse passing through the point $(28,13)$.

$$\begin{gathered} \frac{{28}^2}{a^2}+\frac{{13}^2}{{20}^2}=1 \\ \frac{{28}^2}{a^2}=1-\frac{{13}^2}{{20}^2} \\ \frac{784}{a^2}=1-\frac{169}{400} \\ \frac{784}{a^2}=\frac{231}{400} \\ {a}^2=\frac{400}{231}\times 784 \\ a=\sqrt{\frac{400}{231}\times 784} \\ a=36.845 \end{gathered}$$

Then , the major axis is

$$\begin{gathered} 2a=2(36.845) \\ =73.69 \end{gathered}$$

Therefore, the span of the bridge is $73.69feet$.