Final answer:
The equation for the parabolic arch of a concrete bridge can be determined using the given information. The equation is y = (-1/72)(x-60)² + 50.
Step-by-step explanation:
The equation for the parabolic arch of a concrete bridge can be determined using the given information. We know that the road is 120 feet long and the maximum height of the arch is 50 feet. Since the arch is symmetrical, we can use the vertex form of a parabola equation: y = a(x-h)² + k. The vertex of the parabola is at (h,k) and the x-coordinate of the vertex is the midpoint of the road, which is half of its length. So, h = 120/2 = 60. The y-coordinate of the vertex is the maximum height of the arch, which is 50. Substituting these values into the equation, we get:
y = a(x-60)² + 50
To find the value of a, we need another point on the parabola. Let's choose one of the endpoints of the road. When x = 0, y = 0, because the arch starts at ground level. Substituting these values into the equation, we get:
0 = a(0-60)² + 50
Simplifying, we get 3600a + 50 = 0. Solving for a, we find that a = -50/3600 = -1/72. Substituting this value back into the equation, we get the final equation for the parabolic arch of the concrete bridge:
y = (-1/72)(x-60)² + 50