document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389523>Question 389523</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the rectangular solid of maximum surface area 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 #276059 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=276059\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is actually somewhat similar to a previous solution I posted, in which the question was asking to prove that the rectangular solid of maximum volume inscribed in a sphere was a cube.\r<BR>\n" );
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document.write( "Let x, y, z be the dimensions of the rectangular solid, and 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;\" > (I assigned this number in the previous solution as well). From this, we establish 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;\" > (this follows from Pythagorean theorem)<BR>\n" );
document.write( "Surface area = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28xy+%2B+yz+%2B+zx%29 ALIGN=MIDDLE  ALT=\"2%28xy+%2B+yz+%2B+zx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Such an inscribed cube has side length 1, and has surface area of 6.\r<BR>\n" );
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document.write( "We want to prove that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28xy+%2B+yz+%2B+xz%29+%3C=+6 ALIGN=MIDDLE  ALT=\"2%28xy+%2B+yz+%2B+xz%29+%3C=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%2B+yz+%2B+xz+%3C=+3 ALIGN=MIDDLE  ALT=\"xy+%2B+yz+%2B+xz+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To show this, I used the Cauchy-Schwarz inequality (see below) letting a_1 = x, a_2 = y, a_3 = z, b_1 = y, b_2 = z, b_3 = x.\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%28y%5E2+%2B+z%5E2+%2B+x%5E2%29+%3E=+%28xy+%2B+yz+%2B+zx%29%5E2 ALIGN=MIDDLE   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;\" >, the left hand side equals 9. Therefore we can substitute 9 and take the square root of both sides to obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9+%3E=+%28xy+%2B+yz+%2B+zx%29%5E2 ALIGN=MIDDLE  ALT=\"9+%3E=+%28xy+%2B+yz+%2B+zx%29%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=3+%3E=+xy+%2B+yz+%2B+zx ALIGN=MIDDLE  ALT=\"3+%3E=+xy+%2B+yz+%2B+zx\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "as desired. Note that the equality case occurs when x = y = z, i.e. the rectangular solid is a cube.\r<BR>\n" );
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document.write( "This is probably the easiest solution, there might be other solutions using optimization given a derivative of a function of two variables x,y (since z is determined from x,y). However it would be rather lengthy compared to this solution (it's pretty amazing that the Cauchy-Schwarz inequality produces the result immediately).\r<BR>\n" );
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document.write( "Note: the Cauchy-Schwarz inequality says that, for positive real numbers a_1, a_2, ..., a_n and b_1, b_2, ...b_n, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28a_i%29%5E2%2C+i+=+1%2C+n%29%2Asum%28%28b_i%29%5E2%2C+i+=+1%2C+n%29+%3E=+%28sum%28%28a_i%29%28b_i%29%2C+i+=+1%2C+n%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The equality case occurs when all a_i are equal to c*(b_i) where c is a constant. </SPAN>\n" );
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