We first need to model the curve (find an equation that represents the curve accurately).
A freely hanging cable takes the form of a catenary. The general form for a catenary is the sum of 2 exponential functions:
The Golden Gate Bridge cable is almost a catenary and almost a parabola, but not quite either (because of the weight of the cables, the suspender ropes and the roadway). For the sake of this discussion, we'll assume it is a catenary.
For convenience, we'll place the origin at the lowest point of the cable.
The required curve (after some guess and check) passing through (-640,152), (0,0) and (640,152) is given by:
Here is the graph of the above equation. We can see it pases through the required points.
The derivative of our function is
Using the length of a curve formula, with start point x = -640 and end point x = 640, we have:
`int_-640^640 sqrt[1+(640/663(e^[x//1326]+e^[-x//1326])/(2))^2] dx = 1326.956`
So the length of the central span of the main cable is `1327.0\ "m"`. (You can see the answer in Wolfram|Alpha here.)
Of course, the cable continues on both sides of the towers. The total length of each cable is `2,332\ "m"`.
Easy to understand math videos: