document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=867777>Question 867777</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good afternoon <BR>\n" );
document.write( "The perimeter of a rectangle is 50 yds. What are the dimensions that will produce the maximum area of such a rectangle?\r<BR>\n" );
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document.write( "Thanks :D </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #523187 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=523187\">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( "The maximum area for a rectangle of a given perimeter is when the the rectangle is a square with side measure <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{P}{4}\">.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P\"> represent the perimeter of some arbitrary rectangle.  Then since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P\ =\ 2l\ +\ 2w\">, we can say that the width is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\"> and the length is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{P}{2}\ -\ w\">\r<BR>\n" );
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document.write( "Since area is length times width, we can write an expression for area as a function of width, thus:\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( "which we recognize from algebra as a quadratic in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\">.\r<BR>\n" );
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document.write( "The graph of such a quadratic using <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\"> as the independent variable is a parabola opening downward.  Hence the value of the function at the vertex is the maximum value of the function.  From algebra we know that the horizontal coordinate of the vertex of a parabola described by the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v\ =\ -\frac{b}{2a}\">\r<BR>\n" );
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document.write( "So for the area function developed above:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w_{max}\ =\ -\frac{\frac{P}{2}}{-2}\ =\ \frac{P}{4}\">\r<BR>\n" );
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document.write( "If you achieve the maximum area when the width is one-fourth of the perimeter, then the length must also be one-fourth of the perimeter.  That is, a square with side length one-fourth of the perimeter as claimed.\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( "My calculator said it, I believe it, that settles it<BR>\n" );
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