The cables of a suspension bridge are in the shape of a parabola. The pillars supporting the cable are 600ft. apart and 90.ft above the road. The lowest height of the cable is 10 ft above the road. What is the height of the cable from the road at a point 150 feet from the center?

The situation presented in the question requires the modeling of a cable of suspension as the equation of a parabola. We can find the exact parameters of the equation by using the dimensions provided in the problem.

Let's place the pillars at each side of the line x=0, where the cable reaches its minimum height. The point (0,10) belongs to the curve. Each pillar is located at x=300 ft and x=-300 ft and the height of the cable is 90 ft. It means we have two more points (300,90) and (-300,90). Now we have enough data, we'll use the general formula for a parabola

[tex]y=ax^2+bx+c[/tex]

We only need to find the values of a, b, and c. Let's use the point (0,10)

[tex]10=a.0^2+b\times 0+c[/tex]

Solving for c

[tex]c=10.[/tex]

The equation is now

[tex]y=ax^2+bx+10[/tex]

Using (300,90)

[tex]90=a.300^2+b\times 300+10[/tex]

Rearranging

[tex]80=300^2a+300b[/tex]

Using (-300,90)

[tex]90=a.(-300)^2-b\times 300+10[/tex]

Rearranging

[tex]80=300^2a-300b[/tex]

Subtracting the last two equations

[tex]0=600b[/tex]

[tex]b=0[/tex]

Now we can find the value of a by solving

[tex]80=300^2a[/tex]

[tex]\displaystyle a=\frac{80}{90000}[/tex]

[tex]\displaystyle a=\frac{1}{1125}[/tex]

The model is now complete:

[tex]\displaystyle y=\frac{1}{1125}x^2+10[/tex]

Now we find the height at x=150 ft

[tex]\displaystyle y=\frac{1}{1125}150^2+10[/tex]

[tex]\boxed{y=30\ ft}[/tex]