An arch of the bridge shown in the following figure is semielliptical, and the major axis is horizontal. The arch is w = 40 feet wide and h = 15 feet high. Find the height of the arch d = 5 feet from the center of the base.

Question

Quizplus · Accepted Answer

Using the equation for a semi-ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a=\frac{w}{2}=20$, $b=\frac{h}{2}=7.5$, and the center of the base is $(20,0)$, we can find the equation of the semi-ellipse to be $\frac{(x-20)^2}{20^2}+\frac{y^2}{7.5^2}=1$. Plugging in $x=25$ (since $d=5$ feet away from the center of the base), we can solve for $y$ to get $y \approx 3.367$. Therefore, the height of the arch at $d=5$ feet away from the center of the base is approximately $\boxed{11ec1e8e\_2c19\_7e0d\_be9b\_3363cb611802\_TBX8967\_11}$.

An Arch of the Bridge Shown in the Following Figure