Hyperbolic Cosine curve fitting

[Denyven]

Homework Statement

I need to fit a curve using cosh to a hyperbola with a vertex of (0,0) and a point at (4,7).

The scanned worksheet can be found here
http://img519.imageshack.us/i/scan0001gu.jpg/"
http://img192.imageshack.us/i/scan0002uz.jpg/"

Homework Equations

$$y=a cosh (\frac{x}{a})-a=\frac{a}{2}(e^{\frac{x}{a}}+e^{\frac{-x}{a}})-a$$ "is the formula for a hyperbola at a vertex of 0,0. a is a constant that modifies the shape" that is what the assignment said exactaly

The Attempt at a Solution

I plugged in 4 for x and 7 for y and attempted to solve algebraically, I just got stuck. then I plugged the equation I attempted to solve into wolfram alpha and got the response no integer solution.

http://www.wolframalpha.com/input/?i=7%3D(a/2)(e^(4/a)%2Be^(-4/a))%2Ba+solve+for+a

 

[hgfalling]

The solution value of a isn't an integer.

Try graphing the function

$$y=x cosh (\frac{4}{x})-x-7$$

and use that to help you find the right value of a.

 

[Gigasoft]

There is probably no closed-form solution, and besides, this formula doesn't describe a hyperbola, but a catenary.

 

[Denyven]

The solution value of a isn't an integer.

Try graphing the function

$$y=x cosh (\frac{4}{x})-x-7$$

and use that to help you find the right value of a.

Thanks so much for your help! I graphed that, and since you solved to equation for 0 then turned 0 into y I looked for the x intercept of the graph and the value fit, thank you so much!
But just for the sake of knowing, how can you guys tell if an equation has no closed form solution?