Why Trucks Don’t Hit Power Lines and the Utility of Catenaries to Electric Utilities

Document Type


Publication Date


Publication Title

Demos with Positive Impact


Background: This demo was motivated by a conversation between Lori Braselton and her father, Charles "Nick" Maruth, Jr. [1] of Davenport, Iowa. He posed the problem

Two poles with heights of 100 feet are to be connected by a flexible cable of length 150 feet. How far apart should the poles be placed so that the distance from the bottom of the wire to the ground is 25 feet? HINT: The answer is obvious.

The "obvious" answer and its not so obvious variations are discussed in this demo.

Objective: The purpose of this demo is to illustrate how hyperbolic functions and arc length integrals are used to model hanging cables.

Level: This demo is appropriate for any calculus or advanced mathematical modeling course in which hyperbolic functions and applications of integration to arc length are covered. Hand solutions of the example problems involve inverse functions.

Prerequisites: Students should be familiar with the hyperbolic sine and cosine functions, their derivatives, and arc length integrals.

Platform: Mathematica is used to generate the animations, to symbolically solve equations, and to find values of integrals. Mathematica notebooks are available for download. However, any computer algebra system may be used for symbolic work. The demo is easily adaptable for use with graphing calculators which may be used for numerical work and graphing. Screen shots are included for the calculator solutions of the systems of equations.

Instructor's Notes: The problem of suspending a flexible cable between two poles of equal height is of interest to utility companies that must make sure that power lines suspended over roadways are not subject to being hit by vehicles which pass underneath.