# how long is this cable? arc length question

• Nov 12th 2009, 04:45 PM
superdude
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: $\displaystyle y=10(e^{\frac{x}{20}}+e^{\frac{-x}{20}})$ where -10<=x<=10

how long is the cable?

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

this gives me an impossible (beyond my level) integral to solve
• Nov 12th 2009, 05:06 PM
chisigma
The problem can be easily solved remembering some properties of the hyperbolic functions. In fact is...

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

... so that is...

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Nov 12th 2009, 05:38 PM
superdude
could someone elaborate on a differnt approach? I haven't learned about hyperbolic functions(Worried)