document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=517998>Question 517998</A>: <I><SPAN STYLE=\"word-break: break-all;\"> using 24 feet, what is the largest rectangle area possible? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #345004 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=345004\">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( "Presuming you mean a maximum 24 foot perimeter:\r<BR>\n" );
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document.write( "The rectangle with the greatest area for a given perimeter is a square with sides that measure the perimeter divided by 4.\r<BR>\n" );
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document.write( "The perimeter of a rectangle is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math P = 2l + 2w \ \ \Rightarrow\ \ 2l = P - 2w \ \ \Rightarrow\ \ l = \frac{P}{2} - w\">\r<BR>\n" );
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document.write( "The area of a rectangle is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math A = lw\">\r<BR>\n" );
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document.write( "Substituting from the perimeter equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math A(w) = w\left(\frac{P}{2} - w\right)\ \ \Rightarrow\ \ A(w) = \left(\frac{wP}{2} - w^2\right)\">\r<BR>\n" );
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document.write( "This function graphs to a parabola opening downward meaning that the vertex is a maximum.\r<BR>\n" );
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document.write( "<u>Algebra solution:</u>  The vertex is the maximum, so for a parabola with equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ ax^2\ +\ bx\ +\ c\">, the vertex is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\">.  For this parabola:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{-\frac{P}{2}}{2(-1)},f\left(\frac{-\frac{P}{2}}{2(-1)}\right)\right)\ =\ \left(\frac{P}{4},f\left(\frac{P}{4}\right)\right)\">\r<BR>\n" );
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document.write( "<u>Calculus solution:</u> The maximum value of the function, hence the maximum area, is where the value of the first derivative is equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \frac{d}{dw}A(w)\ =\  -2w + \frac{P}{2}\">\r<BR>\n" );
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document.write( "Set equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math -2w + \frac{P}{2} = 0 \ \ \Rightarrow\ \ 2w = \frac{P}{2} \ \ \Rightarrow\ \ w = \frac{P}{4}\">\r<BR>\n" );
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document.write( "Hence, the maximum area rectangle for a given perimeter is a square with sides of length one-fourth of the perimeter.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-5%2C+16%2C+-10%2C+60%2C%0D%0Agrid%281%29%2C%0D%0Acircle%286%2C%286%29%5E2%2C.2%29%2C%0D%0Agraph%28%0D%0A500%2C+500%2C+-5%2C+16%2C+-10%2C+60%2C%0D%0Ay+=+-x%5E2+%2B+12x%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\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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