Final answer:
To find the equation of the parabola, we can use the vertex form y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Substituting the coordinates (70, 0) and solving for 'a', the equation of the parabola is y = -0.002857x² + 14. To determine the height of the arch 28 meters from the center, substitute x = 28 into the equation and calculate the value of 'y'.
Step-by-step explanation:
To find the equation of the parabola, we can use the vertex form of a parabola which is y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Since the origin is halfway between the arch's feet, the vertex will be at (0, 14). Now we need to find the value of 'a'.
The span of the arch is 140 meters, so the width of the parabola is 140. The height of the arch is 14 meters, which gives us another point on the parabola.
Substituting the coordinates (70, 0) for (x, y) in the equation, we have:
0 = a(70-0)² + 14
0 = 4900a + 14
4900a = -14
a = -14/4900 = -0.002857
Therefore, the equation of the parabola is y = -0.002857x² + 14.
To determine the height of the arch 28 meters from the center, we substitute x = 28 into the equation:
y = -0.002857(28)² + 14
y = -0.002857(784) + 14
y = -2.228 + 14
y = 11.772
Therefore, the height of the arch 28 meters from the center is approximately 11.772 meters.