Question

The main cables of a suspension bridge are ideally parabolic. the cables over a bridge that is 400 feet logn are attached to towers that 100 feet tall. the lower point of the cable is 40 feet abouve the bdrige. find the equation that can be used to model the cables.

Answer 1

Consider the diagram of the problem.

Let the vertex be on the y axis, exactly, at the point (0, 40)

Another point of this parabola is (200, 100), as can be checked from the figure.

The vertex form of the equation of a parabola is :

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

replacing (h, k) with (0, 40), we have:

$y=ax^{2}+40$

to find a, we substitute (x, y) with (200, 100):

$100=a200^{2}+40$

40,000a=60

a=60/40,000=3/(2,000)

So, the equation of the parabola is $y= \frac{3}{2000} x^{2}+40$