Question inspired from Matt's video, [The bubble that break maths](https://www.youtube.com/watch?v=31Om4VrSzb8&t=671s)  

Bubble shape is a cosh function: $\quad y = a \cosh((x-b)/a) $

With symmetry, we can reduce formula to: $ (R/r) = \cosh(H/r) $  

where r = bubble minimum radius, R = bubble maximum radius

To simplify further, let $x = H/r\,,\,k = H/R :$

$x\,/\,k = \cosh(x)\quad\quad→ k = x\,/\cosh(x) $

Bubble will burst when $k > max(x/\cosh(x)) = 0.66274\,34193\,...$  

This limited domain of $x = [0\,, 1.1996\, 78640\,...]$

Solving x from k is trivial, if we use iterations, say, Newton's method.  

But, if we only need 3 digits accuracy, can we have a direct formula ?

Here is my attempt, by fitting a quadratic, and back solve for x  

Points at x = 0.0, 0.4, 1.2:

$k ≈ -0.46590\,x^2 + 1.1114\,x$  

$x ≈ 1.1927 - \sqrt{1.4225 - 2.1464\,k}$

Using Matts video example: $x|\_{k = {0.25\over0.534}} ≈ 0.5465$  

Convert back to bubble minimum radius: $r = {H \over x} ≈ {0.25 \over 0.5465} ≈ 0.4575$ meter

Actual minimum radius is 0.4652 meter, error of 1.7%  

We can fit more points, but solving for k will involve solving for cubic (or higher) roots.

Any idea ?