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Algebra solution containing trig?

  1. Aug 3, 2011 #1
    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

    [tex] x^3 - 15x^2 + 125 = 0 [/tex]

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

    [tex] x = 5 \sqrt{3}\sin\left(\frac{\pi }{9} \right) -5 \cos\left( \frac{\pi }{9} \right) +5 [/tex]

    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.
  2. jcsd
  3. Aug 3, 2011 #2
    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?
  4. Aug 4, 2011 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Try this article:


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