Arc Length of a Curve: Problem Solving

A high-voltage power line spans a 40-meter horizontal distance between two transmission towers, resulting in a 10-meter vertical sag due to the effects of gravity and thermal expansion. The curve formed by the suspended cable is a catenary, which accurately models the behavior of a uniform, flexible cable under its own weight. Unlike a parabolic shape, the catenary is described by the hyperbolic cosine function and offers a precise representation of the cable's form.

In this setup, engineers assign the parameter aaa a value of 20 meters. This parameter corresponds to the ratio between the horizontal tension in the cable and its weight per unit length. By placing the origin at the lowest point of the curve, the towers are located symmetrically at positions 20 meters to either side of the curve.

To determine the exact cable length required for installation, the arc length of the catenary between the two towers is calculated. The formula used is:

The result of this integration is:

This value gives the total length of the power line needed, ensuring proper installation while accounting for the sag caused by the cable’s own weight and thermal effects.