Question

A suspension footbridge has two 2-meter vertical supports 20 meters apart. The lowest point of the cable connecting them is 1 meter above the path. If the y-axis represents the left support what is the vertex form of the equation for the parabola describing the suspension cable?

Answer 1

The final equation is y = 0.01(x - 10)^2 + 1

Explanation:

The vertex form of a parabola is given by f(x) = a(x - b)^2 + c, where (b, c) is the vertex of the parabola.

To find the vertex, it is given that " two 2-meter vertical supports 20 meters apart" and "The lowest point of the cable connecting them is 1 meter above."

The vertex will be at the mid-point between the two supports: b = 20/2 = 10m

It is 1m above so c = 1m

Substituting into f(x), y = f(x) = a(x - 10)^2 + 1

On the support 20m away from y-axis, x=20, y=2

2 = a(20 - 10)^2 + 1

1 = 100a

a = 0.01

The final equation is y = 0.01(x - 10)^2 + 1

Answer 2

y = (x - 10)^2 / 100 + 1

Explanation:

given:

A suspension footbridge has two 2-meter vertical supports 20 meters apart so the cable connecting 'em will be a convex parabola

The lowest point of the cable, which is the vertex, 1 meter above the path.

y-axis represents the left support and the other vertical support is at x=20

by symmetry, the vertex is mid-pt between the two supports, @ x=20/2=10

so vertex is at (10,1)

vertex form of parabola is y = a(x - h)^2 + k, where (h, k) is the vertex

so y = a(x-10)^2 + 1

at y-axis, x=0, y=2

2 = a(0-10)^2 + 1

2 - 1 = 100a

a = 1/100

so the eqn is y = (x - 10)^2 / 100 + 1