document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=997445>Question 997445</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A field bordering a straight stream is to be enclosed and divided in 2. The side of bordering the stream does not need to be fenced.\r<BR>\n" );
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document.write( "If 2400 yards of fencing is to be used, give an equation for the perimeter of the field (middle part must be included).\r<BR>\n" );
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document.write( "Give an equation for the area of the field.\r<BR>\n" );
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document.write( "Rewrite the equation for area so that it contains only one independent variable.\r<BR>\n" );
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document.write( "What are the dimensions of the largest rectangular field that can be enclosed? Find algebraically using the vertex. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #615530 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=615530\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let w = the width<BR>\n" );
document.write( "let l = the length\r<BR>\n" );
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document.write( "perimeter = l + 3w<BR>\n" );
document.write( "the perimeter is equivalent to the amount of fencing required.<BR>\n" );
document.write( "because the property is divided into two pieces, there are 3 widths rather than 2, the third width being the partition in the middle.<BR>\n" );
document.write( "since perimeter = 2400, the formula for perimeter becomes:<BR>\n" );
document.write( "2400 = l + 3w\r<BR>\n" );
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document.write( "the area formula is:<BR>\n" );
document.write( "area = l * w<BR>\n" );
document.write( "this is the area of the enclosure which includes both halves.\r<BR>\n" );
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document.write( "since 2400 = l + 3w, we can solve for l to get l = 2400 - 3w<BR>\n" );
document.write( "formula for area = l * w becomes area = (2400 - 3w) * w<BR>\n" );
document.write( "remove parentheses to get area = 2400w - 3w^2<BR>\n" );
document.write( "this is a quadratic equation.<BR>\n" );
document.write( "rearrange the terms to get area = -3w^2 + 2400w<BR>\n" );
document.write( "set this equation to 0 to get -3w^2 + 2400w = 0<BR>\n" );
document.write( "this is in standard quadratic equation form of aw^2 + bw + c = 0<BR>\n" );
document.write( "a is the coefficient of the w^2 term, b is the coefficient of the w term, c is the constant term.<BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "a = -3<BR>\n" );
document.write( "b = 2400<BR>\n" );
document.write( "c = 0<BR>\n" );
document.write( "the formula for the value of w that provides maximum point on this equation is:<BR>\n" );
document.write( "w = -b/2a<BR>\n" );
document.write( "the maximum point on this equation is the maximum area of the enclosure.<BR>\n" );
document.write( "this results in w = -2400 / -6 which is equal to 400.<BR>\n" );
document.write( "the area is maximized when w = 400.<BR>\n" );
document.write( "when w = 400, the area is equal to 2400w - 3w^2 which becomes 2400*400 - 3*400^2 which results in an area of 480,000 square yards.<BR>\n" );
document.write( "when w = 400, l = 1200 because 2400 = 1200 + 3 * 400.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "perimeter = 2400 = l + 3w<BR>\n" );
document.write( "area = l*w<BR>\n" );
document.write( "area = (2400 - 3w) * w which is equal to 2400w - 3w^2.<BR>\n" );
document.write( "largest area is when w = 400 and l = 1200.<BR>\n" );
document.write( "w is the width of the enclosure.<BR>\n" );
document.write( "l is the length of the enclosure.\r<BR>\n" );
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document.write( "l is the letter L, not to be confused with 1 which is the number one.<BR>\n" );
document.write( "they are unfortunately very close in appearance with the number 1 having a slanted tip on the top while the letter l has a horizontal tip on the top.  <BR>\n" );
document.write( "otherwise they are identical.\r<BR>\n" );
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document.write( "here's a graph of the equation for the area so you can see what the qaution looks like when graphed.\r<BR>\n" );
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document.write( "to graph the equation, the area is represented by y and the width is represented by x.\r<BR>\n" );
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document.write( "it's easy to see from the graph that the maximum area is when x = 400.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/101901.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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