Applications of Hyperbolic Functions
Hyperbolic functions have practical applications, particularly in the modeling of hanging cables. When a cable of uniform density hangs between two supports, it forms a curve known as a catenary. Examples of catenaries include high-voltage power lines, chains hanging between posts, and strands of a spider’s web.

Mathematically, catenaries can be modeled using hyperbolic functions. Specifically, functions of the form y=acosh(xa) represent catenaries. For instance, the graph of y=2cosh(x2) demonstrates this shape effectively.

This visualization helps in understanding how hyperbolic functions apply to real-world structures.
When solving problems related to catenaries and their lengths, we use the arc length formula. This formula helps us determine the length of the hanging cable modeled by a hyperbolic function.
Recall that the formula for arc length is
Assume a hanging cable has the shape 10cosh(x10) for −15≤x≤15, where x is measured in feet. Determine the length of the cable (in feet).