document.write( "Question 1189057: On a sphere of diameter 125cm, two circles of the sphere whose planes are parallel have a radii 23cm and 53cm, respectively. Find the area of the zone included between these circle. \n" ); document.write( "

Algebra.Com's Answer #820315 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "On a sphere of diameter 125cm, two circles of the sphere whose planes are parallel
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\n" ); document.write( "\n" ); document.write( "            This problem has a huge underwater stone, which Edwin did not disclose to you.\r
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\n" ); document.write( "\n" ); document.write( "            There are TWO basic configurations in this problem.\r
\n" ); document.write( "\n" ); document.write( "            One configuration is when the both parallel planes are on one side from the center of the sphere.\r
\n" ); document.write( "\n" ); document.write( "            Another configuration is when the center of the sphere is located BETWEEN the parallel planes.\r
\n" ); document.write( "\n" ); document.write( "            Each configuration produces its own answer.\r
\n" ); document.write( "\n" ); document.write( "            Two basic configurations produce TWO different answers.\r
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document.write( "When both parallel planes are on one side from the center of the sphere, then the distances \r\n" );
document.write( "to the planes from the center of the sphere are\r\n" );
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document.write( "     = 58.114 cm  (rounded)  and   = 31.125 cm  (rounded).\r\n" );
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document.write( "In this configuration, the distance between the planes is the difference \r\n" );
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document.write( "    h =  -  = 58.114 - 31.125 = 24.989 cm  (rounded).\r\n" );
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document.write( "Then the area of the spherical zone between these two circles is\r\n" );
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document.write( "    A =  =  = 9813.15 cm^2.\r\n" );
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document.write( "When the center of the sphere is BETWEEN the parallel planes, then the distances \r\n" );
document.write( "to the planes from the center of the sphere are the same\r\n" );
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document.write( "     = 58.114 cm  (rounded)  and   = 31.125 cm  (rounded).\r\n" );
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document.write( "But in this configuration, the distance between the planes is the SUM \r\n" );
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document.write( "    h =  +  = 58.114 + 31.125 = 89.239 cm  (rounded).\r\n" );
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document.write( "Then the area of the spherical zone between these two circles is\r\n" );
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document.write( "    A = A =  =  = 35044.04375 cm^2.\r\n" );
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document.write( "Thus the problem has TWO answers:  9813.15 cm^2,  when the planes are on one side from the sphere center, \r\n" );
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document.write( "                              and 35044.04 cm^2,  when the sphere center  is between the planes.\r\n" );
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document.write( "    Notice again, that the formula to calculate the surface area of the spherical zone remains the same in both cases\r\n" );
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document.write( "        A = ,\r\n" );
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document.write( "    but the values  \" h \"  of the zone height are DIFFERENT in each case.\r\n" );
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\n" ); document.write( "\n" ); document.write( "For the area of a spherical segment, see these Internet sources\r
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\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Spherical_segment\r
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\n" ); document.write( "\n" ); document.write( "https://mathworld.wolfram.com/Zone.html\r
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\n" ); document.write( "\n" ); document.write( "https://www.math10.com/en/geometry/sphere.html\r
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