1.
(a)
Plot the graph of the function.
Section 3.10 Arc Length and The Golden Gate Bridge Problem
Subsection 3.10.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 3.10.2 Introduction
Arc length is the distance between two points along a section of a curve. If this curve can be represented by a function \(y=f(x)\text{,}\) then we can calculate the length of this curve from \(x=a\) to \(x=b\) with the formula
\begin{equation}
L = \displaystyle\int_{a}^b \sqrt{1 + [f^{\prime}(x)]^2}\, dx\text{.}\tag{3.6}
\end{equation}
In this activity, you will determine the arc length of a variety of functions and curves. You can then apply equation (3.6) above to determine the length of the cable that holds up the Golden Gate Bridge.
Exercises 3.10.3 Exercises
2.
Consider the curve \(y^2 = x^3\text{.}\)
(a)
Plot the graph of the curve using
implicitplot().
(b)
Solve the equation of the curve for \(y\) to get the equations of the top and bottom halves of the curves as functions.
(c)
3.
Consider the function \(x = \frac{1}{3}\sqrt{y}\,(y-3)\text{.}\)
(a)
Plot the graph of the function. Since \(x\) is defined as a function of \(y\text{,}\) you may need to use
implicitplot() to plot the equation of the curve.
(b)
Determine the arc length of this curve for \(1 \leq y \leq 9\text{.}\)
4.
The main span of the Golden Gate Bridge is 1280 metres long, as shown in Figure 3.2. The top of each of the towers is 230 metres above the surface of the water.
Letting the \(x\)-axis be the surface of the water and the \(y\)-axis be at the centre of the bridge, you may assume that the Golden Gate Bridge cable takes the shape of a catenary over the main span with its lowest point at \((0,70)\text{,}\) corresponding to a height of 70 metres above the water.
Aside
The general form for a catenary passing through its lowest point at \((0,k)\) is
\begin{equation}
g(x) = a\left(\cosh\left(\tfrac{x}{a}\right)-1\right)+k\text{,}\tag{3.7}
\end{equation}
or equivalently,
\begin{equation*}
g(x) = a\left(\frac{{ e}^{x/a} + { e}^{-x/a}}{2}-1\right)+k\text{.}
\end{equation*}
This gives a more simplistic model of the main span cable, as shown in Figure 3.3.
In this exercise, you will determine the length of the main span cable (between the two towers).
(a)
Using equation (3.7) and the points provided in Figure 3.3, assign the proper value to \(k\text{.}\)
(b)
Use the coordinates of the top of one of the towers from Figure 3.3, as well as the
fsolve() command to determine the value of \(a\) for this catenary. Assign this value to \(a\text{.}\)
(c)
Using your assigned values of \(k\) and \(a\text{,}\) assign the function in (3.7) and determine the length of the main span cable.
Interesting facts: 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). The actual cable length is 2,332 metres, from shore to shore. The main cables of the Golden Gate Bridge are nearly one metre in diameter (actually, 0.91 metres) and the total length of galvanized steel wire used in both main cables is 129,000 km.
