document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=828224>Question 828224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangular flower garden is 46 m. Find its maximum area.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #499182 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=499182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The general formula for perimeter of a rectangle is:<BR>\n" );
document.write( "P = 2l + 2w<BR>\n" );
document.write( "where \"l\" is the length and \"w\" is the width. For our rectangle the equation is:<BR>\n" );
document.write( "46 = 2l + 2w<br><BR>\n" );
document.write( "Now we will take a moment to solve our perimeter equation for l. Subtracting 2w we get:<BR>\n" );
document.write( "46 - 2w = 2l<BR>\n" );
document.write( "Dividing by 2 we get:<BR>\n" );
document.write( "23 - w = l<br><BR>\n" );
document.write( "The general formula for area of a rectangle is:<BR>\n" );
document.write( "A = l*w<BR>\n" );
document.write( "We solving the perimeter equation for l so we could substitute in for l in the area formula:<BR>\n" );
document.write( "A = (23 - w)*w<BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+23w+-+w%5E2 ALIGN=MIDDLE  ALT=\"A+=+23w+-+w%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The graph of this equation will be a parabola which opens downward (because of the \"-\" in front of the squared term). Its maximum area will be at the vertex of this parabola. So our task is to find the vertex of the parabola.<br><BR>\n" );
document.write( "When a quadratic is in standard form, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F2a ALIGN=MIDDLE  ALT=\"%28-b%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we will put our equation into standard form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-w%5E2+%2B+23w ALIGN=MIDDLE  ALT=\"A+=+-w%5E2+%2B+23w\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with the \"w\" playing the role of \"x\" and the \"A\" being the \"y\". Now we can find the w-coordinate of the vertex:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5Bv%5D=%28-23%29%2F2%28-1%29 ALIGN=MIDDLE  ALT=\"w%5Bv%5D=%28-23%29%2F2%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5Bv%5D=23%2F2 ALIGN=MIDDLE  ALT=\"w%5Bv%5D=23%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the width that creates the maximum area. It is not the maximum area. For that we have to put 23/2 in for the w in:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-w%5E2+%2B+23w ALIGN=MIDDLE  ALT=\"A+=+-w%5E2+%2B+23w\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-%2823%2F2%29%5E2+%2B+23%2823%2F2%29 ALIGN=MIDDLE  ALT=\"A+=+-%2823%2F2%29%5E2+%2B+23%2823%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-%2823%2F2%29%5E2+%2B+23%2823%2F2%29 ALIGN=MIDDLE  ALT=\"A+=+-%2823%2F2%29%5E2+%2B+23%2823%2F2%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+=+-%28529%2F4%29+%2B+%28529%2F2%29 ALIGN=MIDDLE  ALT=\"A+=+-%28529%2F4%29+%2B+%28529%2F2%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+=+-%28529%2F4%29+%2B+%281058%2F4%29 ALIGN=MIDDLE  ALT=\"A+=+-%28529%2F4%29+%2B+%281058%2F4%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+=+529%2F4 ALIGN=MIDDLE  ALT=\"A+=+529%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the maximum possible area is 529/4 square meters. </SPAN>\n" );
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