document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=965885>Question 965885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A group of parents are planning a children’s playground. In order to save on fencing, they are positioning the playground so that one side is immediately adjacent to the school building. If the playground is to be a rectangle 500 square yards in area, what dimensions should they use if they want to use the least amount of fencing?. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590434 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=590434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let l = the length and let w equal the width.\r<BR>\n" );
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document.write( "the area is equal to l*w.\r<BR>\n" );
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document.write( "since the area has to be 500 square feet, then you get 500 = l * w.\r<BR>\n" );
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document.write( "solve for w to get w = 500 / l.\r<BR>\n" );
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document.write( "the perimeter will be equal to l + 2w.\r<BR>\n" );
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document.write( "normally the perimeter is equal to 2l + 2w, but one of the lengths will be the side of the school, so you only need one l.\r<BR>\n" );
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document.write( "since w = 500 / l, the perimeter becomes equal to l + 2 * 500/l\r<BR>\n" );
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document.write( "set y = the perimeter and set x equal the length and the formula for perimeter becomes:\r<BR>\n" );
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document.write( "y = x + 2*500 / x\r<BR>\n" );
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document.write( "this can be simplified to y = x + 1000 / x\r<BR>\n" );
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document.write( "you want to minimize the perimeter which means you want to minimize y.\r<BR>\n" );
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document.write( "if you graph y = x + 1000 / x, you can see where y will be at a minimum.\r<BR>\n" );
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document.write( "the graph looks like this:\r<BR>\n" );
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document.write( "since both x and y have to be greater than 0, you are looking at the right side of the y-axis and above the x-axis on the graph.\r<BR>\n" );
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document.write( "your problem is to find the minimum point on the graph.\r<BR>\n" );
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document.write( "some graphing software will tell you what it is.\r<BR>\n" );
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document.write( "other graphing software will not.\r<BR>\n" );
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document.write( "in the absence of graphing software that will tell you, or a minimum point formula you can use, you need to use calculus to find it.\r<BR>\n" );
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document.write( "the derivative of the equation set to 0 will give you the min/max point of the equation.\r<BR>\n" );
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document.write( "the equation is y = x + 1000 / x\r<BR>\n" );
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document.write( "the derivative with respect to x is y' = 1 - 1000 / x^2\r<BR>\n" );
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document.write( "set this equal to 0 and you get 1 - 1000 / x^2 = 0\r<BR>\n" );
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document.write( "add 1000 / x^2 to both sides of this equation to get 1 = 1000 / x^2\r<BR>\n" );
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document.write( "solve for x^2 to get x^2 = 1000\r<BR>\n" );
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document.write( "solve for x to get x = plus or minus 31.6227766.\r<BR>\n" );
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document.write( "since x has to be positive, then x = 31.6227766.\r<BR>\n" );
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document.write( "since x represents the length, and the area is equal to the length * width, and the area is equal to 500, you get 500 = 31.6227766 * the width.\r<BR>\n" );
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document.write( "solve for the width to get the width = 15.8113883.\r<BR>\n" );
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document.write( "the perimeter is equal to the length plus 2 times the width, so the perimeter will be 31.6227766 + 2 * 15.8113883 which makes the perimeter equal to 63.2455532.\r<BR>\n" );
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document.write( "placing y = 63.2455532 on the grpah will show you that is the minimum point on the graph as shown below:\r<BR>\n" );
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document.write( "the least amount of fencing will be used when the length is equal to 31.62277766 and the width is equal to 15.8113883.\r<BR>\n" );
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document.write( "the area will be 500 square because 31.62277766 * 15.8113883 = 500.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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