When he designed the Gateway Arch in St. Louis, Missouri, architect Eero Saarinen 5
https://en.wikipedia.org/wiki/Eero_Saarinen
wanted to build it in the shape of an inverted catenary. The arch would have a height of \(630\) feet and a width of \(630\) feet. Its cross sections would be equilateral triangles with sides \(54\) feet at ground level shrinking to \(17\) feet at the top. To obtain the shape of the arch, Saarinen decided that the centers of the triangular cross sections should follow the curve
Show that \(y=68.7672\cosh(0.0100333x)\) is not a true catenary.
(b)
Plot the graph of equation (8.15), and determine the \(x\) and \(y\) intercepts.
(c)
Notice that the answers part (b) do not determine an arch whose height and width are exactly \(630\) feet. This is because this curve represents the centers of the triangular cross sections. (Actually the centroids. You will learn about centroids when you take Integral Calculus.)
The sketch below represents a cross–sectional slice of the arch, which is an equilateral triangle. In the sketch each side is equal to \(s\text{,}\) and the point \(P\) is equidistant from \(A\text{,}\)\(B\text{,}\) and \(C\text{.}\) Show that the perpendicular distance from \(P\) to one side is \(\frac{s}{2\sqrt{3}}\text{.}\)
Use the result in part (i) to determine the height and width of the arch.
(d)
The curve Saarinen used is called a weighted catenary. It is the shape of a hanging chain whose density is not uniform. Saarinen decided to go with this since the size of the triangles was decreasing as they approached the apex of the arch. A true catenary arch would be something of the form
Plot the graphs of equation (8.15) and equation (8.16) on the same set of axes and use this to show that they both have the same requisite height and width.
