Question

Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2} .\) Let \(h\) represent the height of the cable from its lowest point to its highest point and let \(2 w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by \(C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} d x\)

Short answer

The length of the cable follows the formula \(C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} dx\), which is derived using the formula for the arc length of a curve and differentiating the function that describes the shape of the cable.

Step-by-step solution

Arc length of a curve

The arc length of a curve given by the equation \(y = f(x)\) from \(x=a\) to \(x=b\) can be calculated using the formula \[C=\int_{a}^{b} \sqrt{1+(f'(x))^{2}} dx\] This rule follows from Pythagoras' theorem applied infinitesimally.

Differentiating the cable function

The function that describes the shape of the cable is given as \(y=kx^{2}\). Its derivative is: \(y'=2kx\). For the maximum height \(h=k w^{2}\), we have \(k=h / w^{2}\). Thus, the derivative becomes \(y' = \frac{2hx}{w^{2}}\).

Calculating cable length

Plugging the derivative into the arc length formula gives \[C=2 \int_{0}^{w} \sqrt{1+(\frac{2hx}{w^{2}})^{2}} dx\] which simplifies to \[C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} dx\]