Solve the problem.
-A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 170 feet and a maximum height of 30 feet. Find the height of the arch at 15 feet from its center.
A)29.1 ft
B)0.2 ft
C)3.7 ft
D)21.3 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 30 feet, and it spans 170 feet, we can assume the vertex of the parabola is at the center of the span, making the coordinate of the vertex (0, 30) since the maximum height is at the center. The parabola crosses the x-axis at $x = \pm 85$ feet (half of the span on either side of the center), which are the roots of the equation, meaning $y = 0$ at these points.To find the equation of the parabola, we use the vertex form of a parabolic equation, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Here, $h = 0$ and $k = 30$, so the equation simplifies to $y = ax^2 + 30$. Since the parabola crosses the x-axis at $x = 85$ and $x = -85$, we can use one of these points to solve for $a$. Plugging $x = 85$ and $y = 0$ into the equation gives us $0 = a(85)^2 + 30$. Solving for $a$ gives us $a = -30/7225$.To find the height of the arch 15 feet from its center, we substitute $x = 15$ into the equation $y = a(x)^2 + 30$, using the $a$ we just found. This gives us $y = -30/7225 \cdot 15^2 + 30$, which calculates to approximately 29.1 feet. Therefore, the height of the arch at 15 feet from its center is approximately 29.1 feet.

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