Question

I NEED HELP PRE-CAL i only have hours left!

The cables of a suspended-deck suspension bridge are in the shape of a parabola. The pillars supporting the cable are 600 feet apart and rise 90 feet above the road. The lowest height of the cable, which is 10 feet above the road, is reached halfway between the pillars. What is the height of the cable from the road at a point 150 feet (horizontally) from the center of the bridge?

Hint: Place the picture on a coordinate plane. You are solving for the y-coordinate of the unknown height.

Answer 1

If you take the point on the bridge directly beneath the lowest point on the cable to be the origin, then the parabola has equation

`y = ax^2+bx+10`

300 ft to either side of the origin, the parabola reaches a value of y = 90, so

`a(300)^2+b(300)+10 = 90 \implies 4500a+15b = 4`

`a(-300)^2+b(-300)+10 = 90 \implies 4500a-15b = 4`

Adding these together eliminates b and lets you solve for a :

`(4500a+15b)+(4500a-15b)=4+4 \\\\ 9000a = 8 \\\\ a = \frac1{1125}`

Solving for b gives

`4500\left(\frac1{1125}\right)+15b=4 \\\\ 4+15b = 4 \\\\ 15b=0 \\\\ b=0`

So the parabola's equation is

`y = (x^2)/(1125)+10`

150 ft away from the origin, the cable is at a height y of

`y = (150^2)/(1125)+10 = \boxed{30}`

ft above the bridge.