document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204973>Question 1204973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular parking lot must have a perimeter of 520 feet and an area of at least 12,000 square feet. Describe the possible lengths of the parking lot. </SPAN>\n" );
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document.write( "x = length<BR>\n" );
document.write( "y = width<BR>\n" );
document.write( "each is in feet\r<BR>\n" );
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document.write( "2(length+width) = perimeter of rectangle<BR>\n" );
document.write( "2(x+y) = perimeter of rectangle<BR>\n" );
document.write( "2(x+y) = 520<BR>\n" );
document.write( "x+y = 520/2<BR>\n" );
document.write( "x+y = 260<BR>\n" );
document.write( "y = -x+260\r<BR>\n" );
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document.write( "length*width = xy = area of the rectangle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+%3E=+12000 ALIGN=MIDDLE  ALT=\"area+%3E=+12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%3E=+12000 ALIGN=MIDDLE  ALT=\"xy+%3E=+12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28-x%2B260%29+%3E=+12000 ALIGN=MIDDLE  ALT=\"x%28-x%2B260%29+%3E=+12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in y = -x+260\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B260x+%3E=+12000 ALIGN=MIDDLE  ALT=\"-x%5E2%2B260x+%3E=+12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B260x-12000+%3E=+0 ALIGN=MIDDLE  ALT=\"-x%5E2%2B260x-12000+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Consider the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B260x-12000+=+0 ALIGN=MIDDLE  ALT=\"-x%5E2%2B260x-12000+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the quadratic formula to solve and you'll find the x intercepts are x = 60 and x = 200\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60+%3C=+x+%3C=+200 ALIGN=MIDDLE  ALT=\"60+%3C=+x+%3C=+200\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then f(x) = -x^2+260x-12000 is nonnegative and leads back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B260x-12000+%3E=+0 ALIGN=MIDDLE  ALT=\"-x%5E2%2B260x-12000+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > being true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%3E=+12000 ALIGN=MIDDLE  ALT=\"xy+%3E=+12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > is only true when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60+%3C=+x+%3C=+200 ALIGN=MIDDLE  ALT=\"60+%3C=+x+%3C=+200\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Due to symmetry, the x coordinate of the vertex is located at the midpoint of those roots. So the x coordinate of the vertex happens when x = (60+200)/2 = 260/2 = 130\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "y = -x+260<BR>\n" );
document.write( "y = -130+260<BR>\n" );
document.write( "y = 130<BR>\n" );
document.write( "and<BR>\n" );
document.write( "xy = 130*130 = 16900\r<BR>\n" );
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document.write( "The largest area possible is 16900 sq ft and it happens when we have a rectangle of dimensions x = 130 and y = 130; i.e. a square with side length 130 ft.\r<BR>\n" );
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document.write( "Another possible parking lot is when x = 80 and y = 180. It produces an area of xy = 80*180 = 14400 sq ft.\r<BR>\n" );
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document.write( "Another possible parking lot is when x = 100 and y = 160. It produces an area of xy = 100*160 = 16000 sq ft.\r<BR>\n" );
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document.write( "There are infinitely many lots we can form. Simply pick any x value in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60+%3C=+x+%3C=+200 ALIGN=MIDDLE  ALT=\"60+%3C=+x+%3C=+200\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find its paired y value. <BR>\n" );
document.write( "Make sure the x and y values add to 260.<BR>\n" );
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