document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=570786>Question 570786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangle is 2012, and lengths of all sides are integers. What is the smallest possible area of this rectangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367787 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=367787\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the dimensions of this rectangle, we know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2By%29=2012 ALIGN=MIDDLE  ALT=\"2%28x%2By%29=2012\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=2012%2F2 ALIGN=MIDDLE  ALT=\"x%2By=2012%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=1006 ALIGN=MIDDLE  ALT=\"x%2By=1006\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1006-x ALIGN=MIDDLE  ALT=\"y=1006-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area, as a function of x is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=%281006-x%29x ALIGN=MIDDLE  ALT=\"A%28x%29=%281006-x%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=-x%5E2%2B1006x ALIGN=MIDDLE  ALT=\"A%28x%29=-x%5E2%2B1006x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That quadratic equation represents a parabola.<BR>\n" );
document.write( "Its axis of symmetry is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1006%2F%282%2A%28-1%29%29 ALIGN=MIDDLE  ALT=\"x=-1006%2F%282%2A%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=503 ALIGN=MIDDLE  ALT=\"x=503\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The maximum area occurs at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=503 ALIGN=MIDDLE  ALT=\"x=503\"   DISABLED_style= \"max-width:90%; height:auto;\" >, when the rectangle is a square.<BR>\n" );
document.write( "Moving away from that point, to either side of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=503 ALIGN=MIDDLE  ALT=\"x=503\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the area decreases.<BR>\n" );
document.write( "Since the length of the sides are integers, the minimum will be for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1005 ALIGN=MIDDLE  ALT=\"x=1005\"   DISABLED_style= \"max-width:90%; height:auto;\" >, when one side measures 1 and the other 1005. It is the same solutionm no matter what side length we call x.<BR>\n" );
document.write( "The minimum area is 1005. </SPAN>\n" );
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