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Math Help - Hyperbolic Functions

  1. #1
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    Question Hyperbolic Functions

    When a particular cable hangs between two poles the shape is a catenary, f(x)=acosh(x/a) where a is the distance from the sag of the cable to the ground. Suppose the poles are 100 feet apart. Find the slope where the cable meets the pole if the sag is 30 feet.


    So from f(x) = a cosh(x/a) I got that f(0) = a and f(50)=acosh(50/a) with 50 being the distance from where the sag is 30 feet to the pole.

    I need to solve for a somehow with acosh(50/a)-a=30... that being the equation that can tell me how far the sag is from the ground... and then i can plug a into f'(x)=sinh(x/a) and f'(50)=sinh(50/a) to get the derivative at the pole.... but I am stuck solving for a.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Hyperbolic Functions

    Are you sure that you are interpreting the problem correctly? If the value of 'a is defined as "the distance from the sag of the cable to the gound," and you are told that "the sag is 30 feet to the pole," perhaps this means that a=30?

    But assuming that your interpretation is correct, then solving acosh(50/a) -a = 30 probably requires the use of a numerical technique.
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  3. #3
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    Re: Hyperbolic Functions

    The sag is 30 feet. As in if a straight line is drawn from one pole to the other, the cable, at its lowest point is 30ft below the straight line.

    My teacher provided that interpretation in class. Im not sure what you mean by numerical technique.....
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  4. #4
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    Re: Hyperbolic Functions

    It seems to be like a tough question instead of discussing it here why don't you take help from experts. There are people who can help you with these kind of problems. You can log on to www.artifextutors.com and get help from there talk to there live chat experts and get help from there.

    Regards
    Jean
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  5. #5
    MHF Contributor ebaines's Avatar
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    Re: Hyperbolic Functions

    Quote Originally Posted by KaylaKelley View Post
    Im not sure what you mean by numerical technique.....
    Trial and error is a brute force numerical technique. If you have access to a spreadsheet you can set the formula up, try a value for a, note whether it's too big or too small, then try anouther value, etc. To get you started - if you start with a = 46 you should be able to iterate to a value for 'a' that is accurate to 4 decimal points pretty quickly. Then use that value for 'a' in your slope equation.
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  6. #6
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    Re: Hyperbolic Functions

    Oh I see. I graphed it in my calculator and was looking for the x intercept but when I took it to my professor its was wrong and my calculator was giving me something different than his. Thank you for the help though.
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  7. #7
    MHF Contributor ebaines's Avatar
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    Re: Hyperbolic Functions

    The catenary shoud look like this:

    Hyperbolic Functions-catenary.jpg
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  8. #8
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    Re: Hyperbolic Functions

    Hi Kayla,
    As was mentioned in an earlier post, solving the equation
    30 + a = a\cosh\left(\frac{50}{a}\right).
    requires the use of a numerical technique.

    There are a number of available methods, you could use the trial and error approach mentioned earlier, or maybe the Bisection method, or perhaps the simple fixed point iteration
    a_{n+1}=a_{n}\cosh\left(\frac{50}{a_{n}}\right)-30.

    Starting with a_{0}=50, this produces the sequence

    47.154, 46.2417, 46.0121, 45.9597, 45.9480, 45.9454, 45.9449, 45.9447, 45.9447.

    Faster convergence would come from the Newton-Raphson method. This produces the iterative formula, (the bottom line of the fraction is the derivative of the top line),

    a_{n+1}= a_{n}- \frac{30+a_{n}-a_{n}\cosh\left(\frac{50}{a_{n}}\right)}{1-\cosh\left(\frac{50}{a_{n}}\right)+\frac{50}{a_{n}  }\sinh\left(\frac{50}{a_{n}}\right)}.

    Again, starting with a_{0}=50, this produces the sequence 45.4977, 45.9392, 45.9447, 45.9447.

    So it would seem that a = 45.9447 (4dp).
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