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A220052
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Decimal expansion of an ellipsoidal cap height, the cap volume being 1/3 of the ellipsoid volume.
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0
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7, 7, 3, 9, 2, 6, 2, 8, 6, 2, 1, 0, 7, 9, 2, 0, 0, 9, 0, 8, 5, 6, 9, 4, 5, 8, 4, 4, 0, 8, 4, 0, 9, 3, 6, 8, 3, 7, 6, 2, 4, 5, 4, 2, 3, 3, 7, 0, 1, 6, 1, 2, 1, 6, 5, 9, 6, 1, 1, 6, 4, 7, 7, 1, 1, 9, 9, 6, 5, 7, 7, 1, 9, 7, 2, 8, 9, 0, 9, 7, 8, 5, 6, 0, 2, 0, 7, 3, 2, 7, 4, 9, 1, 5, 7, 3, 8, 6, 4, 1, 7, 3, 8, 7, 8
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OFFSET
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0,1
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COMMENTS
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Let the equation of the ellipsoid be x^2/a^2 + y^2/b^2 + z^2/c^2 == 1. The volume of the cap cut across the 'a' axis is v(h) = Pi*b*c*(3*a-h)*h^2/(3*a^2). Given v(2a) = 4/3*Pi*a*b*c, and equating v(h) and v(2a)/3, one gets (assuming a=1) the equation 3*h^3 - 9*h^2 + 4 = 0, which is noticeably independant of b or c and valid for a sphere.
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LINKS
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EXAMPLE
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0.773926...
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MATHEMATICA
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RealDigits[N[Root[3*#^3 - 9*#^2 + 4 & , 2], 105]][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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