document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=334094>Question 334094</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A carpenter is building a rectangular room with a fixed perimeter of 312 feet, what dimension would yield the maximum area? What is the maximum area? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #239385 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=239385\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> As a general rule, the maximum area of a rectangle will be the special case when it is a square.<BR>\n" );
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document.write( "Let L=length<BR>\n" );
document.write( "Let W=width<BR>\n" );
document.write( "Let P=perimeter of rectangle<BR>\n" );
document.write( "Let A=area of rectangle<BR>\n" );
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document.write( "we know that P=2*L+2*W<BR>\n" );
document.write( "and that A=L*W<BR>\n" );
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document.write( "Given:  P=312, find L and W such that A is maximum<BR>\n" );
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document.write( "Area: A=L*W<BR>\n" );
document.write( "Perimeter: 312=2*L+2*W<BR>\n" );
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document.write( "312=2*L+2*W<BR>\n" );
document.write( "156=L+W     (divide both sides by 2 to simplify)<BR>\n" );
document.write( "L=156-W      (solve for L)<BR>\n" );
document.write( "A=L*W=(156-W)*W    (substitute for L)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=156%2AW-W%5E2 ALIGN=MIDDLE  ALT=\"A=156%2AW-W%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (distribute W)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-1%2A%28W%5E2-156%2AW%29+ ALIGN=MIDDLE  ALT=\"A=-1%2A%28W%5E2-156%2AW%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (rearrange and factor out -1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-%28w%5E2-156%2AW%2B6084-6084%29 ALIGN=MIDDLE  ALT=\"A=-%28w%5E2-156%2AW%2B6084-6084%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (complete the square by adding and subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28156%2F2%29%5E2=78%5E2=6084 ALIGN=MIDDLE  ALT=\"%28156%2F2%29%5E2=78%5E2=6084\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-%28W-78%29%5E2-6084 ALIGN=MIDDLE  ALT=\"A=-%28W-78%29%5E2-6084\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is a parabola with vertex at (78,-6084)<BR>\n" );
document.write( "Since the leading coefficient of the square term is negative, the curve turns downward, thus this is a maximum.<BR>\n" );
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document.write( "Therefore  W=78 is the maximum. And since L=156-W=156-78=78<BR>\n" );
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document.write( "Answer:  the maximum area is when L=78 and W=78 which is a square.<BR>\n" );
document.write( "  </SPAN>\n" );
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