College Algebra - Parabola Word Problem

Question

The towers of a suspension bridge are 800 feet and rise 160 feet above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the towers.

a. Create an equation that models the cable

b. What is the height of the cable 100 feet from a tower?

I need step by step instructions and the answers please, thank you!

Answer 1

Draw a coordinate system. For the origin, use the point where the cable touches the road, which will be the vertex. The coordinate of the parabola at that point will be (0,0). The coordinates of one tower will be (-400,160) and the other tower (400,160). Now use the vertex form for a parabola:

y = a·(x-h)² + k

where a is a constant and (h.k) is the location of the vertex. We have conveniently located the vertex at (0,0), so h = 0 and k = 0. Now the equation reduces to:

y = a·x²

To finish, you just need the find the value of the constant a. To do that, plug in the coordinates of either tower then solve for a.

For part b, just plug x = 100 into your equation. The y value is the height at that point.