To model this, we are going to set the lower point of the cable as the vertex of our parabola; that vertex will be the origin (0,0), so the line of symmetry of our parabola will be y-axis. Since the lower point of the cable is 220 feet above the see weather, the height of our function will be [tex]748-220=528[/tex]. Also, if the towers are 4200 ft apart, each tower will be half that distance to the line of symmetry of our parabola: [tex] \frac{4200}{2} =2100[/tex]. Now, we can infer that the height of the towers will the y-coordinates of our parabola, whereas the distance from each tower will the x-coordinates; therefore the points [tex](-2100,528)[/tex] and [tex](2100,528)[/tex] are on the graph of the parabola.
Now let use the basic form of the equation of a parabola: [tex]y=ax^{2}[/tex] to find [tex]a[/tex]:
[tex]y=ax^{2}[/tex]
[tex]528=a(2100)^{2}[/tex]
[tex]a= \frac{528}{2100^{2} } [/tex]
Finally, lets replace [tex]a[/tex] in our previous equation to complete our model:
[tex]y=ax^{2}[/tex]
[tex]y= \frac{528}{2100^{2} } x^{2}[/tex]
We can conclude that the model which describes the parabola made by the suspension cables is [tex]y= \frac{528}{2100^{2} } x^{2}[/tex].
