Solve the problem.
-A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 160 feet and a maximum height of 40 feet. Find the height of the arch at 10 feet from its center.
A)2.5 ft
B)39.4 ft
C)0.2 ft
D)5 ft

Question

Quizplus · Accepted Answer

The equation of a parabolic arch can be represented as $y = ax^2 + bx + c$. Given the maximum height of the arch is 40 feet at the center, and the span is 160 feet, we can set up our coordinate system such that the vertex of the parabola (the highest point of the arch) is at the origin (0, 40). This simplifies our equation to $y = ax^2 + 40$, since the vertex form of a parabola is $y = a(x-h)^2 + k$, where (h, k) is the vertex of the parabola. Given the span of 160 feet, the roots of the parabola are at $x = -80$ and $x = 80$, since the parabola is symmetric about the y-axis and the total span is 160 feet. At these points, $y = 0$. Using one of the roots to find $a$, we substitute $x = 80$ and $y = 0$ into the equation:$$0 = a(80)^2 + 40$$$$0 = 6400a + 40$$$$6400a = -40$$$$a = -\frac{40}{6400}$$$$a = -\frac{1}{160}$$So, the equation of the parabola is $y = -\frac{1}{160}x^2 + 40$.To find the height of the arch 10 feet from its center, substitute $x = 10$ into the equation:$$y = -\frac{1}{160}(10)^2 + 40$$$$y = -\frac{1}{160}(100) + 40$$$$y = -\frac{100}{160} + 40$$$$y = -\frac{5}{8} + 40$$$$y = 40 - \frac{5}{8}$$$$y = 39.375$$Rounding to one decimal place, the height of the arch at 10 feet from its center is approximately 39.4 feet.

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