document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019815>Question 1019815</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A Wire Length L Is To Cut Into Two Pieces, One Of Which Is To Be Bent Into The Shape Of A Circle And The Other Into The Shape Of An Equilateral Triangle. Find The Length Of Each Piece So That The Sum Of The Enclosed Areas Is A Minimum<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 #635948 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=635948\">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=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >= radius of the circle<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;\" >= side of the triangle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > length of the wire before cutting it<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2AR ALIGN=MIDDLE  ALT=\"2pi%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of wire used for the circle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=L-2pi%2AR ALIGN=MIDDLE  ALT=\"3x=L-2pi%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of wire used for the triangle\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=L-2pi%2AR ALIGN=MIDDLE  ALT=\"3x=L-2pi%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28L-2pi%2AR%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28L-2pi%2AR%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Area of the triangle =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29x%5E2%2F4=%28sqrt%283%29%2F4%29%28%28L-2pi%2AR%29%2F3%29%5E2=sqrt%283%29%28L-2pi%2AR%29%5E2%2F36 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Area of the circle =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Total area =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=pi%2AR%5E2%2Bsqrt%283%29%28L-2pi%2AR%29%5E2%2F36 ALIGN=MIDDLE  ALT=\"y=pi%2AR%5E2%2Bsqrt%283%29%28L-2pi%2AR%29%5E2%2F36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=pi%2AR%5E2%2Bsqrt%283%29%28L%5E2-4L%2Api%2AR%2B4pi%5E2%2AR%5E2%29%2F36 ALIGN=MIDDLE  ALT=\"y=pi%2AR%5E2%2Bsqrt%283%29%28L%5E2-4L%2Api%2AR%2B4pi%5E2%2AR%5E2%29%2F36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=pi%2AR%5E2%2Bsqrt%283%29L%5E2%2F36-4sqrt%283%29L%2Api%2AR%2F36%2B4sqrt%283%29pi%5E2%2AR%5E2%2F36 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=pi%2AR%5E2%2Bsqrt%283%29L%5E2%2F36-sqrt%283%29L%2Api%2AR%2F9%2Bsqrt%283%29pi%5E2%2AR%5E2%2F9 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28pi%2Bsqrt%283%29pi%5E2%2F9%29%2AR%5E2-%28sqrt%283%29L%2Api%2F9%29%2AR%2Bsqrt%283%29L%5E2%2F36 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is a quadratic function.<BR>\n" );
document.write( "Quadratic functions of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "have a minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F2a ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28pi%2Bsqrt%283%29pi%5E2%2F9%29%2AR%5E2-%28sqrt%283%29L%2Api%2F9%29%2AR%2Bsqrt%283%29L%5E2%2F36 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=R ALIGN=MIDDLE  ALT=\"x=R\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%28pi%2Bsqrt%283%29pi%5E2%2F9%29%3E0 ALIGN=MIDDLE  ALT=\"a=%28pi%2Bsqrt%283%29pi%5E2%2F9%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-sqrt%283%29L%2Api%2F9 ALIGN=MIDDLE  ALT=\"b=-sqrt%283%29L%2Api%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the minimum is at<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=%28sqrt%283%29L%2Api%2F9%29%2F%282%28pi%2Bsqrt%283%29pi%5E2%2F9%29%29=%28sqrt%283%29L%29%2F%282%289%2Bsqrt%283%29pi%29%29=about0.59968 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );