document.write( "Question 1181730: The center of each of three spheres of radius R lies in the surfaces of the other two. Pass a plane containing the centers of the spheres. Find the area common to the three great circles cut from the spheres by this plane.
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Algebra.Com's Answer #850001 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The center of each of three spheres of radius R lies in the surfaces of the other two.
\n" ); document.write( "Pass a plane containing the centers of the spheres.
\n" ); document.write( "Find the area common to the three great circles cut from the spheres by this plane.
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\n" ); document.write( "\n" ); document.write( "        In the post by @CPhill, the solution is INCORRECT conceptually; \r
\n" ); document.write( "\n" ); document.write( "        therefore, his calculations are incorrect and his answer is incorrect, too.\r
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\n" ); document.write( "\n" ); document.write( "        The major fault is that his description and understanding of the common area\r
\n" ); document.write( "\n" ); document.write( "        of intersecting great circles of the spheres is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        See my solution below.\r
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document.write( "The centers of the three great circles form an equilateral triangle.\r\n" );
document.write( "The sides of this triangle are of the length R, since R is the radius of each\r\n" );
document.write( "of the three spheres.\r\n" );
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document.write( "The area of this triangle is  a = .\r\n" );
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document.write( "But the common intersection area is   than this triangle.\r\n" );
document.write( "The common area is formed as the intersection of three   of the great circles,\r\n" );
document.write( "each sector with the central angle of 60°.\r\n" );
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document.write( "So, the common area can be described as the equilateral triangle with the sides length of R\r\n" );
document.write( " three adjacent circular  SEGMENTS  adjacent to each side of this triangle.\r\n" );
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document.write( "The area of each such a segment is  b =  - .\r\n" );
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document.write( "The area of the three such segments is  3b =  - .\r\n" );
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document.write( "Therefore, the total common area of the intersection of the three great circles is\r\n" );
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document.write( "    Area = a + 3b =  +  -  = \r\n" );
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document.write( "                  =  -  =  -  =\r\n" );
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document.write( "                  = .\r\n" );
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document.write( "ANSWER.  The common area of intersection of the three great circles is ,\r\n" );
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document.write( "         or about  0.6998*R^2,  approximately.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Compare it with the answer by @Phill of    $\"sqrt%283%29%2F4%29%2AR%5E2\"$ = 0.433*R^2   (approximately).\r
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @KPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence is like a baby now. It is in the experimental stage
\n" ); document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again, it made an error.\r
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\n" ); document.write( "\n" ); document.write( "Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
\n" ); document.write( "and can make errors/mistakes.\r
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.\r
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\n" ); document.write( "\n" ); document.write( "So, in my view, doing this way, this guy, @CPhill, simply makes dishonest business at this forum.\r
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\n" ); document.write( "\n" ); document.write( "Every time, @CPhill embarrasses to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth, as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it, their reliability is ZERO and their creadability is ZERO, too.\r
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