document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=689122>Question 689122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm having a difficult time with the following problem and would greatly appreciate help.  The problem reads:  I plan to fence my rectangular garden with 100 meters of fencing.  I am using the function A=w(50-w) to express the area in terms of its width.  What is the maximum possible area that I can enclose with this fencing? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #425847 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=425847\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The maximum area of a rectangle always ends<BR>\n" );
document.write( "up being a square when you alter the lengths<BR>\n" );
document.write( "of the sides. I will show this for your problem.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+w%2A%28+50+-+w+%29+ ALIGN=MIDDLE  ALT=\"+A+=+w%2A%28+50+-+w+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+50w+-+w%5E2+ ALIGN=MIDDLE  ALT=\"+A+=+50w+-+w%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "rewrite it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-w%5E2+%2B+50w+ ALIGN=MIDDLE  ALT=\"+A+=+-w%5E2+%2B+50w+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a parabola which has a maximum and<BR>\n" );
document.write( "not a minimum. This is indicated by the minus sign<BR>\n" );
document.write( "in front of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w%5E2+ ALIGN=MIDDLE  ALT=\"+w%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > term<BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "There are 2 equally valid ways to find the w-coordinate<BR>\n" );
document.write( "of the peak ( maximum area ).<BR>\n" );
document.write( "The peak will be midway between the roots which occur<BR>\n" );
document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+0+ ALIGN=MIDDLE  ALT=\"+A+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+w%2A%28+50+-+w+%29+ ALIGN=MIDDLE  ALT=\"+A+=+w%2A%28+50+-+w+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+w%2A%28+50+-+w+%29+ ALIGN=MIDDLE  ALT=\"+0+=+w%2A%28+50+-+w+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This equation is true for 2 values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w+ ALIGN=MIDDLE  ALT=\"+w+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w+=+0+ ALIGN=MIDDLE  ALT=\"+w+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w+=+50+ ALIGN=MIDDLE  ALT=\"+w+=+50+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The peak will be midway between, at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w+=+25+ ALIGN=MIDDLE  ALT=\"+w+=+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------------------------<BR>\n" );
document.write( "The other way to find the maximum is to use the general<BR>\n" );
document.write( "formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Bmax%5D+=+-b%2F%282a%29+ ALIGN=MIDDLE  ALT=\"+x%5Bmax%5D+=+-b%2F%282a%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the equation<BR>\n" );
document.write( "has the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+ax%5E2+%2B+b%2Ax+%2B+c+ ALIGN=MIDDLE  ALT=\"+y+=+ax%5E2+%2B+b%2Ax+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Of course you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w%5Bmax%5D+=+-b%2F%282a%29+ ALIGN=MIDDLE  ALT=\"+w%5Bmax%5D+=+-b%2F%282a%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+a%2Aw%5E2+%2B+b%2Aw+ ALIGN=MIDDLE  ALT=\"+A+=+a%2Aw%5E2+%2B+b%2Aw+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( c = 0 )<BR>\n" );
document.write( "You have <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+-1+ ALIGN=MIDDLE  ALT=\"+a+=+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+50+ ALIGN=MIDDLE  ALT=\"+b+=+50+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w%5Bmax%5D+=+-50%2F%282%28-1%29%29+ ALIGN=MIDDLE  ALT=\"+w%5Bmax%5D+=+-50%2F%282%28-1%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w%5Bmax%5D+=+25+ ALIGN=MIDDLE  ALT=\"+w%5Bmax%5D+=+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Now when you plug <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+w%5Bmax%5D+=+25+ ALIGN=MIDDLE  ALT=\"+w%5Bmax%5D+=+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" > back into your equation, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-+w%5E2+%2B+50w+ ALIGN=MIDDLE  ALT=\"+A+=+-+w%5E2+%2B+50w+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-+25%5E2+%2B+50%2A25+ ALIGN=MIDDLE  ALT=\"+A+=+-+25%5E2+%2B+50%2A25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-625+%2B+1250+ ALIGN=MIDDLE  ALT=\"+A+=+-625+%2B+1250+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+625+ ALIGN=MIDDLE  ALT=\"+A+=+625+\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the maximum area<BR>\n" );
document.write( "Ntice that you have a square because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2AL+%2B+2%2AW+=+C+ ALIGN=MIDDLE  ALT=\"+2%2AL+%2B+2%2AW+=+C+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( formula for circumference )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2A25+%2B+2%2A25+=+C+ ALIGN=MIDDLE  ALT=\"+2%2A25+%2B+2%2A25+=+C+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+50+%2B+50+=+100+ ALIGN=MIDDLE  ALT=\"+50+%2B+50+=+100+\"   DISABLED_style= \"max-width:90%; height:auto;\" > m of fencing<BR>\n" );
document.write( "hope this helps\r<BR>\n" );
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document.write( " </SPAN>\n" );
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