# Math Help - how long is this cable? arc length question

1. ## how long is this cable? arc length question

question:A telephone cable susbended between two poles 20 meters appart hangs in a curve with equation: $y=10(e^{\frac{x}{20}}+e^{\frac{-x}{20}})$ where -10<=x<=10

how long is the cable?

work:using the formula $L=2\int_0^{10} \sqrt{1+(\frac{dy}{dx})^2}dx$

this gives me an impossible (beyond my level) integral to solve

2. The problem can be easily solved remembering some properties of the hyperbolic functions. In fact is...

$y= 20 \cdot \cosh \frac{x}{20}$ (1)

... so that is...

$\frac{dy}{dx}= \sinh \frac{x}{20}$ (2)

$(\frac{dy}{dx})^{2} = \sinh^{2} \frac{x}{20} = \cosh^{2} \frac{x}{20} -1$ (3)

Kind regards

$\chi$ $\sigma$

3. could someone elaborate on a differnt approach? I haven't learned about hyperbolic functions