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.
Figure 3. Chains between these posts take the shape of a catenary. (credit: modification of work by OKFoundryCompany, Flickr)
Mathematically, catenaries can be modeled using hyperbolic functions. Specifically, functions of the form [latex]y=a\text{cosh}\left(\frac{x}{a}\right)[/latex] represent catenaries. For instance, the graph of [latex]y=2\text{cosh}\left(\frac{x}{2}\right)[/latex] demonstrates this shape effectively.
Figure 4. A hyperbolic cosine function forms the shape of a catenary.
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.
Assume a hanging cable has the shape [latex]10\text{cosh}\left(\frac{x}{10}\right)[/latex] for [latex]-15\le x\le 15,[/latex] where [latex]x[/latex] is measured in feet. Determine the length of the cable (in feet).
We have [latex]f(x)=10\text{cosh}(x\text{/}10),[/latex] so [latex]{f}^{\prime }(x)=\text{sinh}(x\text{/}10).[/latex] Then
Watch the following video to see the worked solution to this example.
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