document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=471487>Question 471487</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show that the maximum area of a rectangle inscribed in a circle is a square </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #323618 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=323618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If L and W denote the length and width of a rectangle, and the radius of the circle is r, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE L^2 + W^2 = (2r)^2 = 4r^2\"> (by the Pythagorean theorem)\r<BR>\n" );
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document.write( "Hence, we can say that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE W^2 = 4r^2 - L^2\">, and that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A^2 = L^2W^2 = L^2(4r^2 - L^2)\"> (A denotes the area of the rectangle)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A = \sqrt{L^2(4r^2 - L^2)}\">\r<BR>\n" );
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document.write( "By AM-GM inequality,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{L^2 + (4r^2 - L^2)}{2} \ge \sqrt{L^2(4r^2 - L^2)} = A\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2r^2 \ge A\">\r<BR>\n" );
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document.write( "The left side is a constant, so we can say that the area is maximized in the equality case, which occurs if and only if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE L^2 = 4r^2 - L^2 \Rightarrow L^2 = W^2 \Rightarrow L = W\">\r<BR>\n" );
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document.write( "In other words, the area is maximized when the rectangle is a square. </SPAN>\n" );
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