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Phantom Spheres (Posted on 2012-12-22) Difficulty: 3 of 5

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Some Thoughts Need Mathematica Comment 1 of 1

#1 
   
   Vol D = Vol E
   Vol R - Vol E = Vol E
   Vol R = 2 * Vol E
   (4/3)*pi*R^3 = 2*(4/3)*pi*r^3
   R = r*cuberoot(2)
#2-#4
   Let d be the distance between centers.
   Vol B = Integral from d-r to x_0 of
             [r^2 - (x-d)^2]dx
           + Integral from x_0 to R of
             [R^2 - x^2]dx
           where x_0 = (d^2 + R^2 - r^2)/(2d)
   Vol B = pi*P(d)/(12d) where
           P(d) = d^4 - 6(R^2 + r^2)d^2
                  + 8(R^3 + r^3) - 3(R^2 - r^2)^2
   This checks out:
      If d = R+r, then the spheres are externally
                  tangent and Vol B = 0.
      If d = R-r, then the spheres are internally
                  tangent and Vol B = Vol E.
   For #2, Vol R - Vol E = Vol B
           (4/3)*pi*(R^3 - r^3) = pi*P(d)/(12d)
                       or
           d^4 - 6(R^2 + r^2)d^2 - 8(R^3 - 3r^3) 
               - 3(R^2 - r^2)^2 = 0
   For #3, Vol A = Vol B
           Vol R - Vol B = Vol B
           Vol R = 2 * Vol B
           (4/3)*pi*R^3 = 2*pi*P(d)/(12d)
                       or
           d^4 - 6(R^2 + r^2)d^2 + 8r^3*d 
               - 3(R^2 - r^2)^2 = 0
   For #4, Vol C = Vol B
           Vol E - Vol B = Vol B
           Vol E = 2 * Vol B
           (4/3)*pi*r^3 = 2*pi*P(d)/(12d)
                       or
           d^4 - 6(R^2 + r^2)d^2 + 8R^3*d 
               - 3(R^2 - r^2)^2 = 0
I will let somebody with Mathematica solve the
quartics of #2-#4.

Edited on December 24, 2012, 4:54 pm
  Posted by Bractals on 2012-12-24 16:45:12

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