Chapter 10.3, Problem 76PE

A cable hanging freely between two vertical support beams forms a curve called a catenary. The shape of a catenary resembles a parabola but mathematically the two functions are quite different.

a. On a graphing utility, graph a catenary defined by $y=\frac{1}{2}\left(e^x+e^{-x}\right)$ and graph the parabola defined by $y=x^2+1$ .

b. A catenary and a parabola are so similar in shape that we can often use a parabolic curve to approximate the shape of a catenary. For example, a bridge has cables suspended from a larger approximately parabolic cable. Take the origin at a point on the road directly below the vertex and write an equation of the parabolic cable.

c. Determine the focal length of the parabolic cable.

d. Determine the length of the vertical support cable $100\text{ ft}$ from the vertex. Round to the nearest tenth of a foot.