document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=222959>Question 222959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to figure out different lenghts and widths of a rectangle with perimeters of 500 ft and areas of 21,800 square feet </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #166821 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=166821\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = length and let y = width.\r<BR>\n" );
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document.write( "Your 2 equations are:\r<BR>\n" );
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document.write( "2x + 2y = 500<BR>\n" );
document.write( "xy = 21800\r<BR>\n" );
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document.write( "Solvle for y in both equations to get:\r<BR>\n" );
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document.write( "y = 250-x<BR>\n" );
document.write( "y = 21800/x\r<BR>\n" );
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document.write( "Graph these equations to get what is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1000%2C1000%2C-1000%2C1000%2C250-x%2C21800%2Fx%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1000%2C1000%2C-1000%2C1000%2C250-x%2C21800%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The graphs of these 2 equations do not intersect, therefore you do not have a solution that is common to both.\r<BR>\n" );
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document.write( "If you solve these two equations simultaneously by substitution, it will lead to an equation of x^2 - 250x + 21800 = 0.\r<BR>\n" );
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document.write( "Solve this using the quadratic equation and you will wind up with imaginary roots.\r<BR>\n" );
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document.write( "This means there is no solution for x that is real that will satisfy both equations, I believe.\r<BR>\n" );
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document.write( "The graph of the two equations bears this out.\r<BR>\n" );
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document.write( "I'm not 100% sure that I'm right but it sure looks like that.\r<BR>\n" );
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document.write( "I did a test, however, to see what happens if I do get an intersection on the graph.\r<BR>\n" );
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document.write( "I doubled the perimeter to 1000.\r<BR>\n" );
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document.write( "Your 2 equations would become:\r<BR>\n" );
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document.write( "2x + 2y = 1000<BR>\n" );
document.write( "xy = 21800\r<BR>\n" );
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document.write( "Solvle for y in both equations to get:\r<BR>\n" );
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document.write( "y = 500-x<BR>\n" );
document.write( "y = 21800/x\r<BR>\n" );
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document.write( "Graph these equations to get what is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1000%2C1000%2C-1000%2C1000%2C500-x%2C21800%2Fx%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1000%2C1000%2C-1000%2C1000%2C500-x%2C21800%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You do get an intersection now which says there is a solution to this revised equation.\r<BR>\n" );
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document.write( "Solving it algebraically using the quadratic formula does provide real roots to the equation this time.\r<BR>\n" );
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document.write( "We have a quadratic equation of x^2 - 500x + 21800 = 0 which yields roots of:\r<BR>\n" );
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document.write( "x = 48.25758998<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "x = 451.74241\r<BR>\n" );
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document.write( "When x is 48.25758998, y = 451.74241\r<BR>\n" );
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document.write( "When x is 451.74241, y = 48.25758998\r<BR>\n" );
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document.write( "2 * x + 2 * y = 1000 which is the perimeter of the revised equation.\r<BR>\n" );
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document.write( "x*y = 48... * 451... = 21800 which is the area of the revised equation.\r<BR>\n" );
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document.write( "My original analysis is proved correct because when I do get an intersection ojn the graph, I have real solutions, which means that there is no solution to the equations you posed originally.  Those are:\r<BR>\n" );
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document.write( "2x + 2y = 500<BR>\n" );
document.write( "x*y = 21800\r<BR>\n" );
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document.write( " </SPAN>\n" );
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