document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=196947>Question 196947</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A retangular field is to be enclosed by 500m of fence. What dimensions will give the maximum area? What is the maximum area? Be sure to use \"let statements\" </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147655 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=147655\">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( "Let's solve this one in general, that is for any given length of fence.\r<BR>\n" );
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document.write( "Let <b><i>F</i></b> represent the length of fence available.\r<BR>\n" );
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document.write( "Let <b><i>w</i></b> represent the width of the field.\r<BR>\n" );
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document.write( "Let <b><i>l</i></b> represent the length of the field.\r<BR>\n" );
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document.write( "The perimeter of a rectangle is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P = 2l + 2w\">\r<BR>\n" );
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document.write( "So, in order to enclose the field,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ F = P = 2l + 2w\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ l = \frac{F - 2w}{2}\">\r<BR>\n" );
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document.write( "The area of a rectangle is the length times the width so a function for the area in terms of the width is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w) = \left(\frac{F - 2w}{2}\right)w\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w) = \frac{Fw - 2w^2}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w) = \frac{F}{2}w - w^2\">\r<BR>\n" );
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document.write( "<b><u>Algebra Solution:</u></b>\r<BR>\n" );
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document.write( "The area function is a parabola, opening downward, with vertex at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{-\frac{F}{2}}{-2} = \frac{F}{4}\">\r<BR>\n" );
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document.write( "Since the parabola opens downward, the vertex represents a maximum value of the area function.  The value of the width that gives this maximum value is one-fourth of the available fencing.  Therefore, the shape must be a square, and the area is the width squared.\r<BR>\n" );
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document.write( "<b><u>Calculus Solution:</u></b>\r<BR>\n" );
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document.write( "The area function is continuous and twice differentiable across its domain, therefore there will be a local extrema wherever the first derivative is equal to zero and that extreme point will be a maximum if the second derivative is negative.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w) = \frac{F}{2}w - w^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A'(w) = \frac{F}{2} - 2w\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{F}{2} - 2w = 0 \ \ \Rightarrow\ \ w = \frac{F}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A'' = -2\">\r<BR>\n" );
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document.write( "Therefore the maximum area is obtained when\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w = \frac{F}{4}\">\r<BR>\n" );
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document.write( "And that maximum area is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A\left(\frac{F}{4}\right) = \frac{F}{2}\left(\frac{F}{4}\right) - \left(\frac{F}{4}\right)^2 = \left(\frac{F}{4}\right)^2 = w^2\">\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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