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%I #13 Jan 03 2013 20:49:19
%S 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,
%T 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,
%U 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
%N Decimal expansion of an ellipsoidal cap height, the cap volume being 1/3 of the ellipsoid volume.
%C 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.
%H Eric Weisstein's Mathworld, <a href="http://mathworld.wolfram.com/Ellipsoid.html"> Ellipsoid</a>
%e 0.773926...
%t RealDigits[N[Root[3*#^3 - 9*#^2 + 4 & , 2], 105]][[1]]
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Dec 03 2012
%E Offset changed from 1 to 0 by _Bruno Berselli_, Jan 03 2013
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