Question

The cable of suspension bridge hangs in the shape of a parabola.the towers supporting the cable are 400 ft apart and 150 ft high .if the cable at its lowest is 30 ft above the bridge at its midpoint ,how high is the cable 50 ft away (horizontally) from either tower?

Answer 1

Equation of parabola is $y=ax^2+bx+c$ .

Let coordinates of the cable on one of the towers is (0,150) and on

another (400,150). The coordinates of the middle of the cable will be

(200,30). Then we will have three equations:

$150=a*0^2+b*0+c$

$150=a*400^2+b*400+c$

$30=a*200^2+b*200+c$

From the first equation:

$c=150$

From 2nd and 3rd equations:

$150=a*400^2+b*400+150$

$30=a*200^2+b*200+150$

$a*400^2+b*400=0$

$a*200^2+b*200=-120$

Then we the 2nd equation by 2 and substract from the 1st equation:

$a*(400^2-2*200^2)=240$

$a=240/(400^2-2*200^2)=240/80000=3/1000=0.003$

$b=-a*400=-0.003*400=-1.2$

So equation of parabola is

$y=0.003x^2-1.2x+150$

The height of the cable on the distance 50 feet from tower is

$h=0.003*50^2-1.2*50+150=$

$=0.003*2500-60+150=7.5+90=97.5$