Question

A telephone line hangs between two poles 14 m apart in the shape of the catenary , where and are measured in meters.

(a) Find the slope of this curve where it meets the right pole.
(b) Find the angle between the line and the pol

Answer 1

a) At x=14 the slope will be given by:

`(dy)/(dx)(14)=a\sinh \left({\frac {14-C_(1)}{a}}\right)`.

b) Then, the angle between the line and the pole will be:

`\phi=\pi - \theta`

where
`\theta` is the angle between the tangent to the catenary and the x-axis.

Step-by-step explanation:

The catenary has the following general form:

`y(x)==a\cosh \left({\frac {x-C_(1)}{a}}\right)+C_(2)`

a) The slope at any point will be given by the derivative of y.

`(dy)/(dx)(x)=a\sinh \left({\frac {x-C_(1)}{a}}\right)`

At x=14:

`(dy)/(dx)(14)=a\sinh \left({\frac {14-C_(1)}{a}}\right)`.

b) The angle between the tangent to the catenary and the x-axis at a given point will be given by:

`(dy)/(dx)(x)=tan(\theta)` ⇒
`\theta=tan^(-1) ((dy)/(dx)(x))`

Then, the angle between the line and the pole will be:

`\phi=\pi - \theta`.