document.write( "Question 1190324: Three spherical planets of radius r are on orbits that keep them within viewing distances of one another. At any instant, each planet has a region that cannot be seen from anywhere on the other two planets. What is the total area of the three unseen regions? \n" ); document.write( "

Algebra.Com's Answer #821963 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Three spherical planets of radius r are on orbits that keep them within viewing distances
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document.write( "Let start considering TWO spherical planets. Let the points A and B be their centers.\r\n" );
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document.write( "Will call the planets A and B, using the same letters as their centers.\r\n" );
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document.write( "Consider the straight line AB. Then it is easy to understand what are unseen parts \r\n" );
document.write( "(regions) of each planet, that can not be seen from anythere of the other planet.\r\n" );
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document.write( "This region for planet B is the \"rear\" hemi-sphere of the planet B, which is cut by the plane \r\n" );
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document.write( "Again, the unseen part of the planet B from anythere of planet A is the hemi-sphere \r\n" );
document.write( "of B, cut from B by the great circle at B, which is in the perpendicular plane to line AB.\r\n" );
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document.write( "Same is true for the planet B: the unseen part of A from anythere on the B-surface is \r\n" );
document.write( "the \"rear\" hemi-sphere of A, in relation to the straigh line AB.\r\n" );
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document.write( "It is clear that these simple configurations of the unseen parts are due to the fact \r\n" );
document.write( "that the planets have the same radius.\r\n" );
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document.write( "    |    We completed consideration of two planets and     |\r\n" );
document.write( "    |       now we switch our focus to three planets.      |\r\n" );
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document.write( "Let the planets be A, B and C, named according to their centers.\r\n" );
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document.write( "Whichever disposition of the centers A, B, C would be, their centers A, B and C form \r\n" );
document.write( "some plane in 3D space, and we will consider this plane, assuming that the three planets \r\n" );
document.write( "do not lie in the same straight line. Let \"a\", \"b\" and \"c\" be the interior angles \r\n" );
document.write( "of this triangle, opposite to the sides BC, AC, and AB, respectively.\r\n" );
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document.write( "The part of the planet C, unseen from both A and B, is the \"rear\" part of the C surface between\r\n" );
document.write( "two planes: one plane is perpendicular to line AC and the other plane is perpendicular to line BC.\r\n" );
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document.write( "Since the angle between the lines AC and BC is \"c\", the angle between these two planes is \r\n" );
document.write( "(they are perpendicular to the lines AC and BC).\r\n" );
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document.write( "Therefore, the area of the C surface, which is unseen from A and B is   = .\r\n" );
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document.write( "Similarly, the area of the B surface, which is unseen from A and C is   = \r\n" );
document.write( "and        the area of the A surface, which is unseen from B and C is   = .\r\n" );
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document.write( "To answer the problem question, we should add these three unseen areas, and we will get then\r\n" );
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document.write( "    total area of the three unseen regions =  =  = .\r\n" );
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document.write( "Notice that it is EXACTLY the surface area of one such sphere of the radius \"r\".\r\n" );
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document.write( "ANSWER.  Total area of the three unseen regions is .\r\n" );
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document.write( "CHECK.   You can easily check the answer, by considering three spheres located in the vertices of the EQUILATERAL triangle.\r\n" );
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document.write( "         Then your common sense will tell you that the combined unseen area of the three spheres is exactly the surface area \r\n" );
document.write( "         of one such a sphere.\r\n" );
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