Solve the problem.
-An arch for a bridge over a highway is in the form of a semiellipse. The top of the arch is 35 feet above ground (the major axis). What should the span of the bridge be (the length of its minor axis) if the height 30 feet from the
Center is to be 10 feet above ground?
A) 33.49 ft
B) 117.17 ft
C) 116 ft
D) 66.97 ft

Question

Quizplus · Accepted Answer

The equation of an ellipse centered at the origin with a vertical major axis is given by $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. Given that the top of the arch is 35 feet above the ground, the major axis $2a = 70$ feet (since it's a semiellipse, we consider the full height from top to bottom for the calculation). The height 30 feet from the center being 10 feet above the ground means that when $x = 30$, $y = 25$ (since the total height is 35 feet and we want the height above the ground to be 10 feet, we subtract 10 from 35). Substituting $a = 35$ and the given points into the ellipse equation, we solve for $b$. The span of the bridge, which is the length of the minor axis, is $2b$. Solving for $b$ gives us $b \approx 33.49$, so the span of the bridge is $2b \approx 66.97$ feet.

Solve the Problem. -An Arch for a Bridge Over a Highway Is in Highway