document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=873702>Question 873702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help find two numbers whose sum is 700 and whose product is a maximum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527047 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=527047\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= one of the numbers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=700-x ALIGN=MIDDLE  ALT=\"700-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the other number.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%28700-x%29=700x-x%5E2 ALIGN=MIDDLE  ALT=\"y=x%28700-x%29=700x-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the product of those two numbers.<BR>\n" );
document.write( "That is a quadratic function, and is usually written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%5E2%2B700x ALIGN=MIDDLE  ALT=\"y=-x%5E2%2B700x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The graph is a parabola, looking like this: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-200%2C800%2C-50000%2C150000%2C-x%5E2%2B700x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-200%2C800%2C-50000%2C150000%2C-x%5E2%2B700x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Where is its maximum?<BR>\n" );
document.write( "You may have been shown, and it can be proven that the maximum of a quadratic like<BR>\n" );
document.write( "<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;\" > is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F%222+a%22 ALIGN=MIDDLE  ALT=\"x=-b%2F%222+a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So in this case the maximum happens when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-700%29%2F%282%2A%28-1%29%29=%28-700%29%2F%28-2%29=highlight%28350%29 ALIGN=MIDDLE  ALT=\"x=%28-700%29%2F%282%2A%28-1%29%29=%28-700%29%2F%28-2%29=highlight%28350%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Of course, that means that the other number is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=700-x=700-350=highlight%28350%29 ALIGN=MIDDLE  ALT=\"700-x=700-350=highlight%28350%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is not surprising because it is the same as asking you what rectangle with a half-perimeter of 700 has the greatest area, and we know it is a square.<BR>\n" );
document.write( "The same problem could also be asked as what is the largest rectangular plot that can be fenced when you have 1400 feet of fencing. </SPAN>\n" );
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