Find Distance between Poles Given Hanging Cable

[soroban]

The ends of an 80-foot cable are attached to the tops of two 50-foot pole.
The lowest point of the cable is 10 feet from the ground.
Find the distance between the poles.

                L = 80
         *                 *
         |                 |
         |                 |
         |*               *|
         |                 |
      50 | *             * | 50
         |  *           *  |
         |    *       *    |
         |        *        |
         |        :        |
         |        :10      |
         |        :        |
         *--------+--------*
         : - - -  x  - - - :

The equation of a hanging cable is not a parabola.

It is a catenary, with the basic equation: .$$y \:=\:\frac{e^{ax} + e^{-ax}}{2}$$[There is a reason why this problem is not
. . listed under "Challenge Questions".]

 

[kanderson]

This is what I used to get my bridge design lol. I had to write a paper on this. Wish I could find it on my school computer and give it to you guys. Here is a mathematician from oxford that is inspiring and talks about catenary The Catenary - Mathematics All Around Us. - YouTube Although not an explanation more like trying to get people to see something in mathematics, here is one that explains catenary curve The Catenary - YouTube .

 

[Opalg]

I'm thinking, what happens when the poles get closer and closer together? View attachment 716Aha! So that's why you wrote "pole" rather than "poles", I thought it was just a typo.

 

[soroban]

Hello, Opalg!

You got it!

The word "pole" was indeed a typo.
I've corrected it.

 

[LudusRex]

The distance between the poles can be calculated using the catenary equation and the given information. We know that the lowest point of the cable is 10 feet from the ground, which means that the function will have a y-intercept of 10. We also know that the length of the cable is 80 feet, which means that the distance between the poles is 80 feet minus the combined height of the poles (50 feet + 50 feet).

Using the catenary equation, we can set up the following equation:

10 = (50e^ax + 50e^-ax)/2

Solving for a, we get a = ln(3/5)/50 ≈ -0.0055.

Now, we can use this value of a to find the distance between the poles:

Distance = (ln(3/5) + ln(3/5))/(-0.0055) ≈ 353.6 feet.

Therefore, the distance between the poles is approximately 353.6 feet.