Answer to Question #249721 in Calculus for Cez

Question

1. The cable of a suspension bridge hangs in the shape of a parabola. The towers supporting the cable are 500ft. apart and 200ft. high. If the cable as its lowest is 50ft above the at its midpoint, how high is the cable 100 ft. away (horizontally) from either tower.

Answer 1

Suppose that the equation of the parabola is $y(x)=ax^2+c, a>0.$

The lowest point on the suspended cable is 50 feet above the ground

$$y(0)=a(0)^2+c=50=>c=50$$

The towers are 500 feet apart. The towers are 200 feet high.

$$y(-250)=y(250)=200$$

Then

$$200=a(250)^2+50=>a=\dfrac{150}{62500}=0.0024$$

The equation of the arc is

$$y(x)=0.0024x^2+50$$

$$y(150)=0.0024(150)^2+50=104=y(-150)$$

The height of the cable 100 ft away (horizontally) from either tower is 104 feet.