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:

`y=(a(e^[bx]+e^[-bx]))/(2)`

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:

`y=1280((e^[x//1326]+e^[-x//1326])/(2)-1)`

Here is the graph of the above equation. We can see it pases through the required points.

Model of Golden Gate Bridge cable

The derivative of our function is

`(dy)/(dx)=640/663((e^[x//1326]+e^[-x//1326])/(2))`

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"`.

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