suspension bridge (pre cal word problem)?

Question

a suspension bridge w/ weight uniformly distributed along its length has twin towers that extend 75 m above the road surface and are 400 meters apart. the cables are parabolic in shape and are suspended from the tops of the towers. the cables touch the road surface at the center of the bridge. what is the height of the cables at a point 100 m from the center? (assuming that the road is level)

thanks for the help.

Answer 1

The answer is: 18.75m

Take the center of the bridge (where the parabolic suspension cable touches the bridge) to be the "origin". In this case we have a parabola in the form:

f(x) = (x/a)^2 where "a" is the scaling factor to be determined.

We have: f(200m) = 75m = (200m/a)^2

So a = sqrt(200^2 /75) = 23.09

Now that we know a = 23.09, we can determine the height of the cable 100m from the center. We have:

f(100m) = (x/a)^2 = (100m / 23.09)^2 = 18.75 m

Answer 2

alright, you need to measure from the center because it is a parabola thats symetric along the y axis (why?), so

x =200 meters (from center.)

y = 75 meters (at 200 meters from center)

with relationship

y = A x^2 +B

but B is zero (why?)

solve for A then back substitute 100 meters.