Solve the problem.
-An arch for a bridge over a highway is in the form of a semiellipse. The top of the arch is 30 feet above ground (the major axis). What should the span of the bridge be (the length of its minor axis) if the height 29 feet from the
Center is to be 15 feet above ground?
A) 117.17 ft
B) 116 ft
C) 33.49 ft
D) 66.97 ft

Question

Quizplus · Accepted Answer

The equation for an ellipse centered at the origin is $x^2/a^2 + y^2/b^2 = 1$, where 2a is the length of the major axis and 2b is the length of the minor axis. For a semiellipse that represents the arch, the major axis (height) is 30 feet, so $b = 30/2 = 15$ feet. We're given that at a height of 15 feet above the ground (or 15 feet from the center vertically, since the total height is 30 feet), the distance from the center is 29 feet. This means we're looking for the span when $y = 15$, which is $x = 29$. Plugging into the ellipse equation, we get $x^2/a^2 + y^2/b^2 = 1$, or $29^2/a^2 + 15^2/15^2 = 1$. Solving for $a$ gives us the semi-minor axis, but we need the full span, which is $2a$. The correct calculation involves using the given conditions to solve for the dimensions of the ellipse, and the span of the bridge is twice the value of $a$ found from the conditions given. The correct answer, after solving the equation with the given values, is $66.97$ feet, which corresponds to the total length of the minor axis or the span of the bridge.

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