document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389078>Question 389078</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the rectangular solid of maximum volume that can be inscribed in a sphere is a cube? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #275603 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=275603\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Without loss of generality, let the diameter of the sphere be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (you'll see why I picked such a strange number), and that the dimensions of the rectangular solid are x, y, z. Given that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+%2B+z%5E2+=+%28sqrt%283%29%29%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+%2B+z%5E2+=+%28sqrt%283%29%29%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we want to show that for all positive x, y, z that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xyz+%3C=+1 ALIGN=MIDDLE  ALT=\"xyz+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (where 1 is the equality case and the volume of an inscribed cube).\r<BR>\n" );
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document.write( "Using the AM-GM inequality,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+%2B+y%5E2+%2B+z%5E2%29%2F3+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+y%5E2+%2B+z%5E2%29%2F3+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+%2B+z%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+%2B+z%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29 ALIGN=MIDDLE  ALT=\"1+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Cubing both sides,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3E=+x%5E2y%5E2z%5E2 ALIGN=MIDDLE  ALT=\"1+%3E=+x%5E2y%5E2z%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3E=+xyz ALIGN=MIDDLE  ALT=\"1+%3E=+xyz\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xyz+%3C=+1 ALIGN=MIDDLE  ALT=\"xyz+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as desired. Note that the equality case is where x = y = z = 1, which is also the equality case of AM-GM.\r<BR>\n" );
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document.write( "There is probably an optimization solution involving derivatives, however, it would involve several variables and would be fairly tedious. I'm sure many other solutions exist. This solution is the first one that came to my mind. </SPAN>\n" );
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