Algebra solution containing trig?

[Nebuchadnezza]

I was calculating some numbers, revolving around a sphere and some integration.

The case was to find out for what x-value a spherical bowl with a radius of 5m was half filled.

After doing the algebra, I narrowed it down to

$$x^3 - 15x^2 + 125 = 0$$

Now, my calculator says that the solution to this equation is.

$$x = 5 \sqrt{3}\sin\left(\frac{\pi }{9} \right) -5 \cos\left( \frac{\pi }{9} \right) +5$$

How do you figure that out? Kept trying a few variable changes, but nothing comes to mind.

Ofcourse this polynomial have more solutions, but this is the only solution where 0<x<5 the other ones where -2.2 and 14.

 

[gsal]

where is x? where is the origin? If it was me, I would have placed the origin in the center of the sphere and then for x=0, the sphere should be half full?

 

[SteamKing]

Try this article:

http://en.wikipedia.org/wiki/Cubic_function

There is a section on the trigonometric method for the solution of a cubic eq. toward the bottom of the article.