document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=535681>Question 535681</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose with 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 #351976 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=351976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Be careful. This could be a \"trick\" question. If it is required that the field be a rectangular shape, the maximum area would be enclosed by a field that is square. In that case the field would have all four sides be of length 25 meters. This would give you a perimeter of 100 meters and would enclose a field of dimensions 25 meters by 25 meters, resulting in an enclosed area of 25 times 25 = 625 square meters.<BR>\n" );
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document.write( "But if the farmer is allowed to build a circular fenced area, this will result in a different area being enclosed. The 100 meters of fence would be the circumference of the field. Since we know that the formula for circumference of a circle is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+2%2Api%2AR ALIGN=MIDDLE  ALT=\"C+=+2%2Api%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Where C represents the circumference (in this case C = 100 meters) and R is the radius of the circular field. We can solve for the radius of this field by dividing both sides of the equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api ALIGN=MIDDLE  ALT=\"2%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+C%2F%282%2Api%29 ALIGN=MIDDLE  ALT=\"R+=+C%2F%282%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And substituting 100 for C the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+100%2F%282%2Api%29 ALIGN=MIDDLE  ALT=\"R+=+100%2F%282%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Dividing the numerator of 100 by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api ALIGN=MIDDLE  ALT=\"2%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > results in finding that the radius R is 15.91549431 meters.<BR>\n" );
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document.write( "We also know from geometry that the Area of a circle is given by the formula:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"A+=+pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which A represents the area and R again represents the radius. We know that for this problem R equals 15.91549431 meters. Substitute this value into the equation for the area and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+pi%2A%2815.91549431%29%5E2 ALIGN=MIDDLE  ALT=\"A+=+pi%2A%2815.91549431%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Square the radius and multiply that result by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you find that the circular field contains an area of 795.7747154 square meters. <BR>\n" );
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document.write( "This is more area than if the field were a square of 25 meters by 25 meters. The area made by fencing a circle from 100 meters of fence contains nearly 171 square meters more (795.7747154 square meters minus 625 square meters = 170.7747154 square meters). <BR>\n" );
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document.write( "Hope this helps you understand the problem a little better.<BR>\n" );
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