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 #590531 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=590531\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C220%2C-18%2C18%2C-17.9%2C1.9%2C%0D%0Ared%28rectangle%28-15.8%2C0%2C15.8%2C-15.8%29%29%2C%0D%0Aline%28-18%2C0%2C18%2C0%29%2Clocate%28-9%2C1.4%2Cbuilding%29%2C%0D%0Alocate%28-1%2C0%2CL-2x%29%2Clocate%2815.9%2C-7.5%2Cx%29%2C%0D%0Alocate%28-1%2C-15.8%2CL-2x%29%2Clocate%28-16.5%2C-7.5%2Cx%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" >= total length of fence (in yards), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >= area in square yards,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of each side of the fence directly attached to the building<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you are studying Calculus, you would want to use functions and derivatives to find the maximum of Area as a function of one of the measurements.\r<BR>\n" );
document.write( "\n" );
document.write( "If you are not studying calculus, or learning how to use a graphing calculator, you need an easier solution:<BR>\n" );
document.write( "The more common version of this type of problem, asks for the maximum area for a given total length of fence.<BR>\n" );
document.write( "The general answer is that the best enclosure is always a double square,<BR>\n" );
document.write( "meaning that the rectangular enclosure looks like two squares attached to the wall:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C220%2C-18%2C18%2C-17.9%2C1.9%2C%0D%0Agreen%28rectangle%28-15.8%2C0%2C0%2C-15.8%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C15.8%2C-15.8%29%29%2C%0D%0Aline%28-18%2C0%2C18%2C0%29%2Clocate%28-9%2C1.4%2Cbuilding%29%2C%0D%0Alocate%28-10%2C0%2Cx%29%2Clocate%2815.9%2C-7.5%2Cx%29%2C%0D%0Alocate%289%2C0%2Cx%29%2Clocate%28-16.5%2C-7.5%2Cx%29%2C%0D%0Alocate%289%2C-15.8%2Cx%29%2Clocate%280%2C-7.5%2Cx%29%2Clocate%28-10%2C-15.8%2Cx%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In other words, half of the length of the fence is the long side parallel to the building wall, <BR>\n" );
document.write( "the other half is used for the two sections of fence that connect that long side to the wall.<BR>\n" );
document.write( "To prove that, you would write that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=x%28L-2x%29 ALIGN=MIDDLE  ALT=\"A=x%28L-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-2x%5E2%2BLx ALIGN=MIDDLE  ALT=\"A=-2x%5E2%2BLx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function that graphs as a parabola,<BR>\n" );
document.write( "and has a vertex/maximum for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-L%2F%282%28-2%29%29 ALIGN=MIDDLE  ALT=\"x=-L%2F%282%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=L%2F4 ALIGN=MIDDLE  ALT=\"x=L%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28L=4x%2CL-2x=2x%29 ALIGN=MIDDLE  ALT=\"system%28L=4x%2CL-2x=2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives you the most efficient way to fence three sides of an enclosure,<BR>\n" );
document.write( "yielding the maximum area for a given total length of fence:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=%28L-2x%29%28x%29=%282x%29%28x%29=2x%5E2=2%28L%2F4%29%5E2=L%5E2%2F8 ALIGN=MIDDLE  ALT=\"A%5Bmax%5D=%28L-2x%29%28x%29=%282x%29%28x%29=2x%5E2=2%28L%2F4%29%5E2=L%5E2%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%3C=L%5E2%2F8 ALIGN=MIDDLE  ALT=\"A%3C=L%5E2%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Obviously, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D ALIGN=MIDDLE  ALT=\"A%5Bmax%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > (because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3E0 ALIGN=MIDDLE  ALT=\"L%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , of course).<BR>\n" );
document.write( "So, if they just use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40 ALIGN=MIDDLE  ALT=\"40\"   DISABLED_style= \"max-width:90%; height:auto;\" > yards of fencing material, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=40 ALIGN=MIDDLE  ALT=\"L=40\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the largest rectangular are you can enclose is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200 ALIGN=MIDDLE  ALT=\"200\"   DISABLED_style= \"max-width:90%; height:auto;\" > square yards:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=40%5E2%2F8=1600%2F8=200 ALIGN=MIDDLE  ALT=\"A%5Bmax%5D=40%5E2%2F8=1600%2F8=200\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "They will need more fencing material.<BR>\n" );
document.write( "If the parents use  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" > yards of fencing material, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=60 ALIGN=MIDDLE  ALT=\"L=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the largest rectangular are you can enclose is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=450 ALIGN=MIDDLE  ALT=\"450\"   DISABLED_style= \"max-width:90%; height:auto;\" > square yards:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=60%5E2%2F8=3600%2F8=450 ALIGN=MIDDLE  ALT=\"A%5Bmax%5D=60%5E2%2F8=3600%2F8=450\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "They will need more fencing material.<BR>\n" );
document.write( "What total length of fencing material will they need?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%3C=L%5E2%2F8 ALIGN=MIDDLE  ALT=\"A%3C=L%5E2%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8A%3C=L%5E2 ALIGN=MIDDLE  ALT=\"8A%3C=L%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288A%29%3C=L ALIGN=MIDDLE  ALT=\"sqrt%288A%29%3C=L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%3E=2sqrt%282A%29 ALIGN=MIDDLE  ALT=\"L%3E=2sqrt%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For an area of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=450 ALIGN=MIDDLE  ALT=\"A=450\"   DISABLED_style= \"max-width:90%; height:auto;\" > square yards, they will need at least<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=2sqrt%282%2A500%29=2sqrt%281000%29=about+63.25 ALIGN=MIDDLE  ALT=\"L=2sqrt%282%2A500%29=2sqrt%281000%29=about+63.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > yards of fencing material,<BR>\n" );
document.write( "and the dimensions of the enclosure will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=L%2F4=2sqrt%281000%29%2F4=sqrt%281000%29%2F2=about ALIGN=MIDDLE  ALT=\"x=L%2F4=2sqrt%281000%29%2F4=sqrt%281000%29%2F2=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815.81%29 ALIGN=MIDDLE  ALT=\"highlight%2815.81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > yards,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L-2x=2x=2%28sqrt%281000%29%2F2%29=sqrt%281000%29=about ALIGN=MIDDLE  ALT=\"L-2x=2x=2%28sqrt%281000%29%2F2%29=sqrt%281000%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2831.62%29 ALIGN=MIDDLE  ALT=\"highlight%2831.62%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > yards.<BR>\n" );
document.write( " <BR>\n" );
document.write( "IF YOU ARE STUDYING CALCULUS:<BR>\n" );
document.write( "You would write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of some measure,<BR>\n" );
document.write( "and calculate the derivative.<BR>\n" );
document.write( "For example, if we were to use as a variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the length of the parallel fence sides,<BR>\n" );
document.write( "the length of the other side would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Fx ALIGN=MIDDLE  ALT=\"500%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the total length of fencing needed would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B500%2Fx ALIGN=MIDDLE  ALT=\"y=2x%2B500%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The derivative of that functions is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=2-500%2Fx%5E2 ALIGN=MIDDLE  ALT=\"dy%2Fdx=2-500%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=%282x%5E2-500%29%2Fx%5E2 ALIGN=MIDDLE  ALT=\"dy%2Fdx=%282x%5E2-500%29%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That derivative is zero when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-500=0 ALIGN=MIDDLE  ALT=\"2x%5E2-500=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=sqrt%28250%29%2C%22%27or%22%2Cx=-sqrt%28250%29%29 ALIGN=MIDDLE  ALT=\"system%28x=sqrt%28250%29%2C%22%27or%22%2Cx=-sqrt%28250%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The derivative is negative (and the function decreases) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%28250%29%3Cx%3Csqrt%28250%29 ALIGN=MIDDLE  ALT=\"-sqrt%28250%29%3Cx%3Csqrt%28250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Esqrt%28250%29 ALIGN=MIDDLE  ALT=\"x%3Esqrt%28250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the derivative is positive and the function increases,<BR>\n" );
document.write( "so the function has a maximum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28250%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15.81 ALIGN=MIDDLE  ALT=\"15.81\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the lengths of two of the sides of the fence.<BR>\n" );
document.write( "The length of the other side of the fence is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Fx=500%2Fsqrt%28250%29=2sqrt%28250%29 ALIGN=MIDDLE  ALT=\"500%2Fx=500%2Fsqrt%28250%29=2sqrt%28250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31.62 ALIGN=MIDDLE  ALT=\"31.62\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );