document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185698>Question 1185698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> being the first grandson, your grandparents decided to give you a rectangular field for your coming wedding. If you are given 200m wires of fencing, what dimensions would you choose to get the maximum area?<BR>\n" );
document.write( "a. List all the possible dimensions of the rectangular field.<BR>\n" );
document.write( "b. Make a table of values for the possible dimensions.<BR>\n" );
document.write( "c. Compute the area for each possible dimension.<BR>\n" );
document.write( "d. What is the maximum area you obtained?<BR>\n" );
document.write( "e. What are the dimensions of the maximum area you obtained? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849625 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this optimization problem:\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "We have a fixed perimeter (200m of fencing) and want to maximize the area of a rectangle.\r<BR>\n" );
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document.write( "**a. Possible Dimensions:**\r<BR>\n" );
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document.write( "Let *l* be the length and *w* be the width of the rectangle.  The perimeter is given by:\r<BR>\n" );
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document.write( "2*l + 2*w = 200\r<BR>\n" );
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document.write( "We can simplify this to:\r<BR>\n" );
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document.write( "l + w = 100\r<BR>\n" );
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document.write( "Since length and width must be positive, the possible dimensions are any values of *l* and *w* that satisfy this equation, where 0 < l < 100 and 0 < w < 100.  It's impossible to list *all* possible dimensions because they are continuous, not discrete, but we can list some examples:\r<BR>\n" );
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document.write( "*   l = 10, w = 90<BR>\n" );
document.write( "*   l = 20, w = 80<BR>\n" );
document.write( "*   l = 30, w = 70<BR>\n" );
document.write( "*   l = 40, w = 60<BR>\n" );
document.write( "*   l = 50, w = 50<BR>\n" );
document.write( "*   l = 60, w = 40<BR>\n" );
document.write( "*   l = 70, w = 30<BR>\n" );
document.write( "*   l = 80, w = 20<BR>\n" );
document.write( "*   l = 90, w = 10\r<BR>\n" );
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document.write( "**b. Table of Values:**\r<BR>\n" );
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document.write( "It's helpful to organize the dimensions and areas in a table. Here are some examples:\r<BR>\n" );
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document.write( "| Length (l) | Width (w) | Area (A) |<BR>\n" );
document.write( "|---|---|---|<BR>\n" );
document.write( "| 10 | 90 | 900 |<BR>\n" );
document.write( "| 20 | 80 | 1600 |<BR>\n" );
document.write( "| 30 | 70 | 2100 |<BR>\n" );
document.write( "| 40 | 60 | 2400 |<BR>\n" );
document.write( "| 50 | 50 | 2500 |<BR>\n" );
document.write( "| 60 | 40 | 2400 |<BR>\n" );
document.write( "| 70 | 30 | 2100 |<BR>\n" );
document.write( "| 80 | 20 | 1600 |<BR>\n" );
document.write( "| 90 | 10 | 900 |\r<BR>\n" );
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document.write( "**c. Computing the Area:**\r<BR>\n" );
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document.write( "The area of a rectangle is given by:\r<BR>\n" );
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document.write( "A = l * w\r<BR>\n" );
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document.write( "We've already calculated the areas in the table above.\r<BR>\n" );
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document.write( "**d. Maximum Area:**\r<BR>\n" );
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document.write( "From the table, we can see that the maximum area is 2500 m².\r<BR>\n" );
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document.write( "**e. Dimensions of Maximum Area:**\r<BR>\n" );
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document.write( "The dimensions that give the maximum area are l = 50m and w = 50m.  This means the rectangle with the maximum area is a square.\r<BR>\n" );
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document.write( "**General Solution (using calculus):**\r<BR>\n" );
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document.write( "While the table gives the answer, calculus can be used for a more general approach.\r<BR>\n" );
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document.write( "1.  Express *w* in terms of *l*:  w = 100 - l\r<BR>\n" );
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document.write( "2.  Area formula: A = l(100 - l) = 100l - l²\r<BR>\n" );
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document.write( "3.  To find the maximum area, take the derivative of A with respect to *l* and set it equal to zero:\r<BR>\n" );
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document.write( "    dA/dl = 100 - 2l = 0<BR>\n" );
document.write( "    2l = 100<BR>\n" );
document.write( "    l = 50\r<BR>\n" );
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document.write( "4.  Since w = 100 - l, then w = 100 - 50 = 50.\r<BR>\n" );
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document.write( "This confirms that the maximum area occurs when l = 50 and w = 50.<BR>\n" );
document.write( " </SPAN>\n" );
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