document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90369>Question 90369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you wanted to construct a fence around a garden plot in the form of a rectangle. On the neighbor’s side it’s going to need heavy-duty fencing that costs $2.00 per foot. The other three sides can be made of standard fencing material that costs $1.20 foot. You have $200 to spend. <BR>\n" );
document.write( "1. How many different rectangles would it be possible to enclose for $200? <BR>\n" );
document.write( "2. Write an equation using two variables for the total cost of the fence. Use x and y to represent the width and length of the rectangle. Solve the equation for one of the variables (either x or y). <BR>\n" );
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document.write( "3. Given that Area = (x)(y), now write the area function for this problem using ONLY ONE VARIABLE (by making a substitution using the equation from #2). <BR>\n" );
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document.write( "4. What dimensions will give the maximum area inside the fence? <BR>\n" );
document.write( "5. What will be the maximum area? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65578 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=65578\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.\r<BR>\n" );
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document.write( "First draw the picture. In this drawing the thicker line at the top denotes the neighbor's side (which is denoted as N)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0A+++++++++locate%280%2C2.5%2CN%29%2C%0D%0A+++++++++locate%280%2C-2.5%2CW%29%2C%0D%0A+++++++++blue%28line%28-3%2C1.98%2C3%2C1.98%29%29%2C%0D%0A+++++++++blue%28line%28-3%2C1.99%2C3%2C1.99%29%29%2C%0D%0A+++++++++blue%28line%28-3%2C2%2C3%2C2%29%29%2C%0D%0A+++++++++blue%28line%28-3%2C2.01%2C3%2C2.01%29%29%2C%0D%0A+++++++++blue%28line%28-3%2C2.02%2C3%2C2.02%29%29%2C%0D%0A%0D%0A+++++++++locate%28-3.5%2C0%2CL%29%2C%0D%0A%0D%0A+++++++++locate%283.5%2C0%2CL%29%2C%0D%0A%0D%0A+++++++++red%28line%283%2C2%2C3%2C-2%29%29%2C%0D%0A+++++++++blue%28line%283%2C-2%2C-3%2C-2%29%29%2C%0D%0A+++++++++red%28line%28-3%2C-2%2C-3%2C2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice the length of W is equal to N\r<BR>\n" );
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document.write( "So the perimeter of this rectangle is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=N%2BW%2B2L ALIGN=MIDDLE  ALT=\"P=N%2BW%2B2L\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And the cost function is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2N%2B1.2W%2B1.2%2A2L ALIGN=MIDDLE  ALT=\"C=2N%2B1.2W%2B1.2%2A2L\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now since N=W, we can replace N with W\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2W%2B1.2W%2B1.2%2A2L ALIGN=MIDDLE  ALT=\"C=2W%2B1.2W%2B1.2%2A2L\"   DISABLED_style= \"max-width:90%; height:auto;\" > replace N with W\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=3.2W%2B1.2%2A2L ALIGN=MIDDLE  ALT=\"C=3.2W%2B1.2%2A2L\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=3.2W%2B2.4L ALIGN=MIDDLE  ALT=\"C=3.2W%2B2.4L\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 1.2 and 2L to get 2.4L\r<BR>\n" );
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document.write( "Also, since you have at most is $200, this means the cost of the fence <b>must</b> be less than $200. So we have this inequality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%3C200 ALIGN=MIDDLE  ALT=\"C%3C200\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now replace C with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.2W%2B2.4L ALIGN=MIDDLE  ALT=\"3.2W%2B2.4L\"   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=3.2W%2B2.4L%3C200 ALIGN=MIDDLE  ALT=\"3.2W%2B2.4L%3C200\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets solve for L\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.4L%3C200-3.4W ALIGN=MIDDLE  ALT=\"2.4L%3C200-3.4W\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3.4W from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C%28200-3.4W%29%2F2.4 ALIGN=MIDDLE  ALT=\"L%3C%28200-3.4W%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C83.33-1.41667W ALIGN=MIDDLE  ALT=\"L%3C83.33-1.41667W\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide \r<BR>\n" );
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document.write( "Now since we don't know the value of W, we cannot solve for L. Since there are two unknown variables, there are an infinite number of possible rectangles. For instance, the neighbor's side might be 10 feet\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C83.33-1.41667%2810%29 ALIGN=MIDDLE  ALT=\"L%3C83.33-1.41667%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in w=10\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C69.1633 ALIGN=MIDDLE  ALT=\"L%3C69.1633\"   DISABLED_style= \"max-width:90%; height:auto;\" > evaluate\r<BR>\n" );
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document.write( "So in this case there are 69 possible rectangles (if the length is a whole number)\r<BR>\n" );
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document.write( "or \r<BR>\n" );
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document.write( "the neighbor's side might be 20 feet\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C83.33-1.41667%2820%29 ALIGN=MIDDLE  ALT=\"L%3C83.33-1.41667%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in w=20\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3C54.9966 ALIGN=MIDDLE  ALT=\"L%3C54.9966\"   DISABLED_style= \"max-width:90%; height:auto;\" > evaluate\r<BR>\n" );
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document.write( "So in this case there are 53 possible rectangles (if the length is a whole number)\r<BR>\n" );
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document.write( "If the length can be any decimal number, then there are an infinite number of rectangles\r<BR>\n" );
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document.write( "This list goes on...\r<BR>\n" );
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document.write( "So this means we cannot figure out how many possible combinations of rectangles we can make. So I would double check your problem. If that is the correct info, then there's a strong chance that there are an infinite number of rectangles.\r<BR>\n" );
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document.write( "------------------------------------------------------------------------\r<BR>\n" );
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document.write( "2. \r<BR>\n" );
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document.write( "We previously solved for the cost function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=3.2W%2B2.4L ALIGN=MIDDLE  ALT=\"C=3.2W%2B2.4L\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Just replace W and L with x and y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=3.2x%2B2.4y ALIGN=MIDDLE  ALT=\"C=3.2x%2B2.4y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets solve for y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=3.2x%2B2.4y ALIGN=MIDDLE  ALT=\"C=3.2x%2B2.4y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the cost function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C-3.2x=2.4y ALIGN=MIDDLE  ALT=\"C-3.2x=2.4y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3.2x from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28C-3.2x%29%2F2.4=y ALIGN=MIDDLE  ALT=\"%28C-3.2x%29%2F2.4=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.4\r<BR>\n" );
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document.write( "So we get\r<BR>\n" );
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document.write( "------------------------------------------------------------------------\r<BR>\n" );
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document.write( "3.\r<BR>\n" );
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document.write( "Area:<BR>\n" );
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document.write( "Using our previous answer, plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28C-3.2x%29%2F2.4 ALIGN=MIDDLE  ALT=\"y=%28C-3.2x%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=xy ALIGN=MIDDLE  ALT=\"A=xy\"   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=A=x%28%28C-3.2x%29%2F2.4%29 ALIGN=MIDDLE  ALT=\"A=x%28%28C-3.2x%29%2F2.4%29\"   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=A=%28Cx-3.2x%5E2%29%2F2.4 ALIGN=MIDDLE  ALT=\"A=%28Cx-3.2x%5E2%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "Since the cost must be what we have in our pocket, the cost is 200. So c=200\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28200x-3.2x%5E2%29%2F2.4 ALIGN=MIDDLE  ALT=\"A=%28200x-3.2x%5E2%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in c=200\r<BR>\n" );
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document.write( "------------------------------------------------------------------------\r<BR>\n" );
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document.write( "4.\r<BR>\n" );
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document.write( "If we plot <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28200x-3.2x%5E2%29%2F2.4 ALIGN=MIDDLE  ALT=\"A=%28200x-3.2x%5E2%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of x, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+100%2C+-10%2C+2000%2C+%28200x-3.2x%5E2%29%2F2.4%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+100%2C+-10%2C+2000%2C+%28200x-3.2x%5E2%29%2F2.4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice how the graph peaks at a certain value. At this x-value, this x-value is one dimension which forms the maximum area. \r<BR>\n" );
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document.write( "So using a calculator, we get x=41.667\r<BR>\n" );
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document.write( "This means that the maximum area is formed when one dimension is 41.667\r<BR>\n" );
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document.write( "Now plug in x=41.667 into  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28200-3.2x%29%2F2.4 ALIGN=MIDDLE  ALT=\"y=%28200-3.2x%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the other dimension\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28200-3.2%2841.667%29%29%2F2.4 ALIGN=MIDDLE  ALT=\"y=%28200-3.2%2841.667%29%29%2F2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And with a calculator we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=27.777 ALIGN=MIDDLE  ALT=\"y=27.777\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the maximum area is formed when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=41.667 ALIGN=MIDDLE  ALT=\"x=41.667\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=27.777 ALIGN=MIDDLE  ALT=\"y=27.777\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------------------------------------------------------------------------\r<BR>\n" );
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document.write( "5.\r<BR>\n" );
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document.write( "Now plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=41.667 ALIGN=MIDDLE  ALT=\"x=41.667\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=27.777 ALIGN=MIDDLE  ALT=\"y=27.777\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=xy ALIGN=MIDDLE  ALT=\"A=xy\"   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=A=%2841.667%29%2827.777%29 ALIGN=MIDDLE  ALT=\"A=%2841.667%29%2827.777%29\"   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=A=1157.384259 ALIGN=MIDDLE  ALT=\"A=1157.384259\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the maximum area is 1,157.384259 square feet </SPAN>\n" );
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