document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1146109>Question 1146109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You want to form a rectangular pen of area a = 70 square feet. (See the figure below.) One side of the pen is to be formed by an existing building and the other three sides by a fence. If w is the length, in feet, of the sides of the rectangle perpendicular to the building, then the length of the side parallel to the building is 70/w, so the total amount F = F(w), in feet, of fence required is the rational function<BR>\n" );
document.write( "F = 2w + <BR>\n" );
document.write( "70<BR>\n" );
document.write( "w<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "Determine the dimensions of the rectangle that requires a minimum amount of fence. (Round your answers to two decimal places.)<BR>\n" );
document.write( "width	    	<BR>\n" );
document.write( " ft<BR>\n" );
document.write( "length	    	<BR>\n" );
document.write( " ft<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #767399 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=767399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The function for the length of fence is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%28w%29+=+2w%2B70%2Fw ALIGN=MIDDLE  ALT=\"F%28w%29+=+2w%2B70%2Fw\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Find the value of x that minimizes the area by finding where the derivative is zero.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dF%2Fdw+=+2-70%2Fw%5E2 ALIGN=MIDDLE  ALT=\"dF%2Fdw+=+2-70%2Fw%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-70%2Fw%5E2+=+0 ALIGN=MIDDLE  ALT=\"2-70%2Fw%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=2+=+70%2Fw%5E2 ALIGN=MIDDLE  ALT=\"2+=+70%2Fw%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2w%5E2+=+70 ALIGN=MIDDLE  ALT=\"2w%5E2+=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2+=+35 ALIGN=MIDDLE  ALT=\"w%5E2+=+35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+sqrt%2835%29 ALIGN=MIDDLE  ALT=\"w+=+sqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The width that minimizes the total amount of fence required is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2835%29 ALIGN=MIDDLE  ALT=\"sqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet.<br><BR>\n" );
document.write( "That makes the length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70%2Fsqrt%2835%29+=+%282%2A35%29%2Fsqrt%2835%29+=+2%2Asqrt%2835%29 ALIGN=MIDDLE  ALT=\"70%2Fsqrt%2835%29+=+%282%2A35%29%2Fsqrt%2835%29+=+2%2Asqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note that, to minimize the total length of fence required, the length is twice the width.  That is always the case, regardless of what the total area is.<br><BR>\n" );
document.write( "So if you are in a position where you see this kind of problem often -- e.g. you are on a high school math team -- then you can just memorize this fact.<br><BR>\n" );
document.write( "Then to solve this problem without doing the calculus, you just solve<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w%29%282w%29+=+70 ALIGN=MIDDLE  ALT=\"%28w%29%282w%29+=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "leading very quickly to dimensions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2835%29 ALIGN=MIDDLE  ALT=\"sqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%2835%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );