Exercise on Parabola with Vertex at (h,k)

Question

The cable of a suspension bridge hangs in the form of a parabola. The distance between the two posts is 150 meters and the height of each post is 22 meters high. If the lowest point in the cable is 7 meters above the bridge, find the vertical distance to the cable from a point in the roadway 15 meters from the foot of a tower/post.

Answer 1

I'm guessing the picture looks like this:

This is a bit confusing because the posts are said to be 22 meters tall but I'm assuming they are 22 meters above the roadway. So the vertex, based on the x, y coordinate plane as shown, is at (0, 7) and the top of the bridge posts are at the coordinates (-75, 22) and (75, 22)

Using the vertex form of the quadratic equation, y = a(x - h)² + k and plugging in the vertex, we get:

y = a(x - 0)² + 7 or

y = ax² + 7

Plugging in the point (75, 22) for x and y allows us to solve for "a":

22 = a(75²) + 7

15 = 5625a

a = 1/375

So y = (1/375)x² + 7

To find the height (y) at the point in question, we plug in x = (75 - 15) or x = 60

y = (1/375)(60²) + 7 = 16.6 meters