document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=738857>Question 738857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you wanted to build area for your dog. You have 24 meters of fencing,each in 1 meter sections. What rectangular shape would produce the largest area for your dog? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #450950 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=450950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "You are in luck.  The given number of fence panels is divisble by 4 which means that it is possible to maximize the area without compromising so that the panels fit.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P\"> represent the available perimeter, i.e. the total length of fencing that you have.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE l\"> represent the length of your rectangle, and let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\"> represent the width of the rectangle.\r<BR>\n" );
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document.write( "Since perimeter, length, and width are related thusly:\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( "we can define length in terms of width and perimeter thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ l\ =\ \frac{P}{2}\ -\ w\">\r<BR>\n" );
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document.write( "Since Area is length times width, using the above expression for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE l\"> we can write a function that yields area as a function of width.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(w)\ =\ -w^2\ +\ \frac{P}{2}w\">\r<BR>\n" );
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document.write( "The astute student should recognize this function as a quadratic in standard form with coefficients <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\ =\ -1\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\ =\ \frac{P}{2}\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c\ =\ 0\">.  You should also note that, given the negative lead coefficient, the parabolic graph opens downwards meaning that the value of the function at the vertex is a maximum.\r<BR>\n" );
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document.write( "Using the formula for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\">-coordinate of the vertex:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w_v\ =\ \frac{-b}{2a}\ =\ \frac{-\frac{P}{2}}{2(-1)}\ =\ \frac{P}{4}\">.\r<BR>\n" );
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document.write( "So the width, in terms of the available perimeter, that yields the greatest area is the perimeter divided by 4.  That means that 2 times the width is the perimeter divided by 2.  Subtracting the perimeter divided by 2 from the perimeter, that leaves the perimeter divided by 2 which represents 2 times the length.  Hence, the length of the maximum area rectangle is also <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{P}{4}\">.\r<BR>\n" );
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document.write( "Therefore, for a given perimeter, the maximum area rectangle that can be constructed is a square with sides that measure one fourth of the perimeter.\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" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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