At work I was facing the problem of how to calculate the height of a water column inside an horizontal cylinder given the volume of the liquid. A plot of this function and a visual explanation can be found here:
http://mathworld.wolfram.com/HorizontalCylindricalSegment.html
I do know this problem can only be solved numerically and do so right now since the function can not be solved for x:
$$f(x)=\frac{1}{\pi}\left(\arccos(1-x)-(1-x)\sqrt{1-(1-x)^2}\right)$$ Here f is the fill fraction ranging from 0 to 1 and x the Ratio of h/R ranging from 0 to 2. And these properties
$f'(0)=0 \quad$ $f'(1)=\frac{2}{\pi}\quad$ $f'(2)=0\quad$ and $\quad f(0)=0 \quad$ $f(1)=\frac{1}{2}\quad$ $f(2)=1$ And of course is monotonic. But since this seems like a basic property of a circle and the resulting function looks so clean I was wondering if there is a type of function that has this shape and could be tuned to calculate a good estimate using it.
So I guess my question is: How can I calculate a good estimate for x without a basic taylor series?