document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316673>Question 316673</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One principle used by the ancient Greeks to get shapes that are pleasing to the eye in art and architecture was the Golden Rectangle. If a square is removed from one end of a Golden Rectangle, the sides of the remaining rectangle are proportional to the original rectangle. So the length and width of the original rectangle satisfy L/W=W/L-W. An artist wants her paining to be in the shape of a golden rectangle. If the length of the painting is 36 inches, then what should be the width? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226587 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=226587\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The ratio of the width to the length of a golden rectangle is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ =\ \frac{1\ +\ \sqrt{5}}{2}\ \approx\ 1.618\">.  In other words:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ l\ =\ w\varphi\">\r<BR>\n" );
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document.write( "but since we are given the length, we need to solve\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w\ =\ \frac{l}{\varphi}\">\r<BR>\n" );
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document.write( "The exact answer to the question can be found relatively easily by noting that the golden ratio, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\">, is unique among positive numbers in that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{\varphi}\ =\ \varphi\ -\ 1\"> and since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ =\ \frac{1\ +\ \sqrt{5}}{2}\">, we can say that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \varphi\ -\ 1\ =\ \frac{-1\ +\ \sqrt{5}}{2}\"> (verification left to the student)\r<BR>\n" );
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document.write( "Hence, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large l\ =\ 36\"> then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w\ =\ 18\left(-1\ +\ \sqrt{5}\right)\text{ in}\">\r<BR>\n" );
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document.write( "Which is the simplest form of the exact answer.\r<BR>\n" );
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document.write( "However, from a practical point of view, what is wanted is a dimension, to the nearest inch of an approximation of the width for a golden rectangle with a length of 36 inches.\r<BR>\n" );
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document.write( "Just multiply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 36 \times 0.62\"> and round to the nearest inch.  Why to the nearest inch?  Because your only measurement was given to the nearest inch and you should never have the result of a calculation based on a measurement expressed to a greater precision than the least precise of your input measurements.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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