Calculus - Catenary - Cable Suspension Bridge - Analysis

ACME Construction Company is building a suspension bridge over the Miami River. They need to know how much material will be required to construct the main support cables and what sort of cable they need to buy. The support cables will be attached at either end to the top of 100 meter tall concrete pillars. The two concrete pillars are 200 meters apart. The cable should hang down 50 meters at its lowest point. Gottfried Leibniz and Christian Huygens in 1691 determined that any cable hanging under the force of gravity must have the shape of the graph

This shape is known as a catenary. The parameter "a" is the ratio of cable tension to cable density and . The only use of the parameter b is to provide a vertical shift, if necessary. ACME would like to hire your group to find two things for them. First, what values must a and b have in order for the catenary to fit the constraints imposed by the placement of the concrete pillars and the low point of the cable? They are especially interested in the parameter a since this tells them what tension the cable will be under. Second, what length cable do they need? You should try to give a formula for the cable length in terms of the cable function y(x). That way, ACME can use your result for other cable shapes as well.
Following are several hints for solving this problem. When you write your report for ACME, you must explain to them, step by step, how you solved the problems. You will need to use a combination of graphs, equations and text. You must try to convince them that your results are correct. It's no use to them if they have to solve the problem themselves in order to verify your results.