 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Phantom Spheres (Posted on 2012-12-22) No Solution Yet Submitted by brianjn No Rating Comments: ( Back to comment list | You must be logged in to post comments.) 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 = 0I will let somebody with Mathematica solve thequartics of #2-#4.`

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

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