document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112897>Question 112897</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular garden with an area of 2050 square feet is to be located next to a barn with fencing on three sides and the barn acting as the fourth side. Write a function which gives the length of fencing needed if the length of the side parellel to the barn is x. Find the dimensions of the garden using the minimum amount of fence. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82269 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82269\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the side parallel to the barn is x and the area is 2050, then the two sides perpendicular to the barn are each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2050%2Fx ALIGN=MIDDLE  ALT=\"2050%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let L(x) be the length function and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29+=+x ALIGN=MIDDLE  ALT=\"L%28x%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Let W(x) be the width function and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%28x%29=2050%2Fx ALIGN=MIDDLE  ALT=\"W%28x%29=2050%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let P(x) be the perimeter function representing the length of the three fenced sides, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=L%28x%29%2B2W%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29=L%28x%29%2B2W%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x) will have a critical point where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%28x%29%2Fdx+=+0 ALIGN=MIDDLE  ALT=\"dP%28x%29%2Fdx+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%28x%29%2Fdx=dL%28x%29%2Fdx%2B2dW%28x%29%2Fdx ALIGN=MIDDLE  ALT=\"dP%28x%29%2Fdx=dL%28x%29%2Fdx%2B2dW%28x%29%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%28x%29%2Fdx=1-2%2A2050%2Fx%5E2 ALIGN=MIDDLE  ALT=\"dP%28x%29%2Fdx=1-2%2A2050%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2%2A2050%2Fx%5E2=0 ALIGN=MIDDLE  ALT=\"1-2%2A2050%2Fx%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4100=0 ALIGN=MIDDLE  ALT=\"x%5E2-4100=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%284100%29 ALIGN=MIDDLE  ALT=\"x=sqrt%284100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now the question is whether this critical point is a minimum or a maximum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2P%28sqrt%284100%29%29%2Fdx%5E2%3E0 ALIGN=MIDDLE  ALT=\"d%5E2P%28sqrt%284100%29%29%2Fdx%5E2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the critical point is a minimum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2P%28x%29%2Fdx%5E2=2%2A4100%2Fx%5E3 ALIGN=MIDDLE  ALT=\"d%5E2P%28x%29%2Fdx%5E2=2%2A4100%2Fx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2P%28sqrt%284100%29%29%2Fdx%5E2=2%2A4100%2F%28sqrt%284100%29%29%5E3%3E0 ALIGN=MIDDLE  ALT=\"d%5E2P%28sqrt%284100%29%29%2Fdx%5E2=2%2A4100%2F%28sqrt%284100%29%29%5E3%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the critical point is a minimum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284100%29 ALIGN=MIDDLE  ALT=\"sqrt%284100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the length x that gives the minimum length fence.  The sides are then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2050%2Fsqrt%284100%29 ALIGN=MIDDLE  ALT=\"2050%2Fsqrt%284100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284100%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%284100%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > when you rationalize the denominator.  I would submit the answer in this form since these values are exact.  If you want to do the calculator work, you will get x to be a little larger than 64 (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64%5E2=4096 ALIGN=MIDDLE  ALT=\"64%5E2=4096\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the sides will be a little more than 32, giving you a bit more than 128 feet of fencing required.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope this helps,<BR>\n" );
document.write( "John  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );