Solve the problem.
-A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 158 feet and a maximumheight of 30 feet. Find the height of the arch at 10 feet from its center.
A)29.5 ft
B)0.1 ft
C)1.9 ft
D)8.8 ft

Question

Quizplus · Accepted Answer

The equation of a parabolic arch can be represented as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Given the maximum height of the arch is 30 feet, this means $k = 30$. The span of the bridge is 158 feet, so the roots of the parabola are at $x = -79$ and $x = 79$ (since the center of the bridge is the vertex of the parabola, and the span is symmetric about the center). The vertex is thus at $(0, 30)$.To find $a$, we use one of the roots, say $x = 79$, and the fact that at this point $y = 0$ (the height at the end of the bridge is 0):$$0 = a(79)^2 + 30$$$$a = -\frac{30}{79^2}$$To find the height at 10 feet from the center, we substitute $x = 10$ into the equation:$$y = -\frac{30}{79^2}(10)^2 + 30$$$$y = -\frac{30}{79^2} \cdot 100 + 30$$$$y = -\frac{3000}{79^2} + 30$$Calculating the exact value:$$y \approx 29.5$$Therefore, the height of the arch at 10 feet from its center is approximately 29.5 feet.

Solve the Problem. -A Bridge Is Built in the Shape of a Parabolic