The first cables on a suspension bridge are connected 50 ft above the roadbed at the ends of the bridge and 10 feet above the roadbed at the ends of the bridge and 10 ft above it in the center of the bridge. The roadbed is 200ft long. Additional vertical cables are to be spaced 20 feet along the bridge. Calculate the minimum amount of cable needed to complete both sides of the bridge.
The cable connecting the sides will show a parabolic curve. The important points to be considered are (0, 50), (100, 10), and (200, 50). The equation is:
y= 0.004x^2 -0.8x + 50
*See attached image
Every 20 meters, a rope is attached vertically, so the points are:
(0,50), (20, 35.6), (40, 24.4), (60, 16.4), (80, 11.6), (100, 10). The rest of the values repeat from of the assumption that the parabola is symmetrical. There are different methods in determining the length of the arc line/parabolic line. Substitute the values for each ordered pairs plotted in the parabola with the formula: sqrt((x2 - x1)^2 + (y2 - y1)^2). The distances are: 24.64, 22.92, 21.54, 20.57 and 20.06. The total distance is the twice the sum of the results or 219.5 ft. We also need to add the vertical lines that are attached every 20 ft in the bridge. (50 + 35.6 + 24.4 + 16.4 + 11.6)*2 + 10 = 286 ft. (Note: The 10 foot cable only occurs once on each side not double like the rest of the values)
Finally, add the total distances of the arc and the vertical ropes.
219.5 + 286 = 505.5 ft.