[Solved] Figure 6.4.20 shows a cable for a suspension bridge. The cable has the shape of a parabola with equation y=k x^2. The suspension bridge has total


Question: Figure 6.4.20 shows a cable for a suspension bridge. The cable has the shape of a parabola with equation \(y=k x^{2}\). The suspension bridge has total span \(2S\) and the height of the cable (relative to its lowest point) is \(H\) at each end. Show that the total length of the cable is given by

\[L=2 \int_{0}^{S} \sqrt{1+\frac{4 H^{2}}{S^{4}} x^{2}} d x\]

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