Question

An arch is in the shape of a parabola. it has a span of 140 meters and a maximum height of 14 meters. arches nat park: landscape arch wayne stadler cc-by-nc-nd find the equation of the parabola (assuming the origin is halfway between the arch's feet).

Answer 1

The equation of the parabolic arch with a 140-meter span and 14-meter height, with the vertex at the origin, is
`\rm y = -1/350x^2 + 14`

Step-by-step explanation:

To find the equation of the parabola that models the arch with a span of 140 meters and a maximum height of 14 meters, we can use the vertex form of a parabolic equation
`\rm y = a(x-h)^2 + k`, where (h, k) is the vertex of the parabola. Since the origin is set at halfway between the arch's feet, the vertex is at (0, 14). The span of the arch indicates that it crosses the x-axis at points (70, 0) and (-70, 0).

To find the value of 'a', we use one of the x-intercepts (70, 0):

`\rm 0 = a(70 - 0)^2 + 14`
0 = 4900a + 14
-14 = 4900a
a = -14 / 4900
a = -1/350

Therefore, the equation of the parabola is y =
`\rm -1/350x^2 + 14.`

Answer 2

The equation of the parabola modeling the arch with a span of 140 meters and a maximum height of 14 meters, with its vertex at the origin, is y = -0.00285714286x^2 + 14.

Step-by-step explanation:

The student is asking for the equation of a parabola that models an arch with a span of 140 meters and a height of 14 meters, assuming the vertex of the parabola is at the origin. To find the equation of the parabola, we can use the standard form y = ax2 + bx + c, where the vertex is at (0, c), which in this case is (0, 14), and the parabola opens downwards. Since the arch is 140 meters wide and symmetric, the x-intercepts are located at (-70,0) and (70,0).

Considering the vertex form y = a(x - h)2 + k, with (h,k) being the vertex and since the vertex is at the origin (0,14), the equation simplifies to y = ax2 + 14. Next, we substitute one of the x-intercepts into the equation to solve for 'a'. Using (70,0):

0 = a(70)2 + 14

a = -14 / (70)2

a = -0.00285714286

Thus, the equation of the parabola is y = -0.00285714286x2 + 14.