Suspension bridge

Question

A suspension bridge has twin towers that extend 75 meters 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 tough the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (assume the road is flat) also, How many feet away from the center will the cable be exactly 3 meters from the road surface?

Answer 1

The graph of a parabola can be described by an equation of the form y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

 

Orient the x and y axes so that the origin is on the roadway at the center of the bridge.  Then the cable is parabolic with vertex (0, 0) and with equation y = ax².  Since (200, 75) is a point on the cable, we have:

 

75 = a(200)²     So, a = 75/(200)²  = 0.001875

 

Therefore, y = 0.001875x²

 

The height of the cable 100 meters from the center = 0.001875(100)² = 18.75 meters.

 

When the height of the cable is 3 meters, 3 = 0.001875x²

 

                                                               x² = 1600

 

                                                               x = 40  meters