The Arc length of a cable hanging from two poles x feet apart.

[violin_writer]

Homework Statement

I appologize if this is in the wrong topic. But, I need help with the. I know you guys don't exactly give out the answer, but I'm looking for a particular rule of something that will help me. My calculus professor told me to use any available resource to solve this problem. The question is in blue.

Find the arc length of a steel cable hanging between two 50 feet tall poles that are 1000 feet apart and the cable is 15 feet above the ground at its lowest point. Hint: use the hyperbolic cosine centenary to find a, then use the sinh equation derived in class, round to the nearest foot.


I have a specific problem. Being told to use any available resource, gave me idea to turn to my calculator. Well I keep getting a number that appears to be too high for any of the answers.

Homework Equations

I these r the equations... y= a(cosh((x/a)-1)+M. I use to find a.
And the arc length formula is 2asinh(x/a)

Where:
x= the distance a pole is from its origin =500
y= the pole height= 50
m= the height of the cable at its origin = 15
a= 95.269...

And ofcourse

The Attempt at a Solution

The following answer is wrong because it is way too large. Could someone show me what I did wrong?

 

[vela]

Your answer for a is wrong. When I plug it into your equation for y, I don't get an answer anywhere near close to 50.

 

[violin_writer]

Ok so I looked it over... could it y is wrong. That that is an invalid equation? Is what I started with not the "hyperbolic centernary equation"?

Thanks 4 helping... but I need more help

 

[vela]

Regardless of whether the equation for y is right or wrong, if you solve for a when y=50, you should find y=50 when you plug your value for a back into the equation. The fact that doesn't happen doesn't mean the equation is wrong; it means you solved for a incorrectly. How exactly did you solve for a?

By the way, your original equation for y and the one you show in your attempt don't match. It might just be a typo. I tried plugging your value for a into both versions, and it doesn't work for either.

 

[violin_writer]

I used solver on my calculator (ti-84)... it's the only way we've gone over. For some reason it either gets overflowed, or iteration errors. The only way that seems to work is by giving a 95 for a.

Basically I did what was shown... I set the equation equal to zero & used the "solver" command on my calc... then the last equation to get arc length. I did it pretty much how he did it. What makes me think typo, is the huge difference n the numbers (like the thousand's really 100 or something). And the fact that I graphed the equation and it never touches 50.
I was hoping someone on here would catch something I wasn't doing or come up with an extra step I need to actually get the arc length.

And which one isn't matching? the Sinh one with an extra parenthesis, that is a typo... thanks.

 

[vela]

Oh, I read your original equation to be

$$y = a (\cosh\left(\frac{x}{a}\right) - 1) + M$$

because you wrote it with an extra parenthesis. That's actually the correct equation. The way you're using it

[tex]y = a \cosh\left(\frac{x}{a}-1\right) + M[/itex]

isn't correct. You can see this by plugging x=0 in. It should reduce to y=M.

 

[violin_writer]

thanks dude that was it... right there. It's funny how something small like that can effect something.