document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199429>Question 199429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A length of wire is cut into two pieces, one of which is bent to form a circle, the other a square.  How should the wire be cut if the area enclosed by the two curves is to be a minimum? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149879 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=149879\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the total length of the wire = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let the length of a side of the square = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the square will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the square will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x ALIGN=MIDDLE  ALT=\"4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the circle will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+4x ALIGN=MIDDLE  ALT=\"1+-+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let the perimeter of the circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let the radius = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2%2Api%2Ar ALIGN=MIDDLE  ALT=\"P+=+2%2Api%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2Ar+=+1+-+4x ALIGN=MIDDLE  ALT=\"2%2Api%2Ar+=+1+-+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%281+-+4x%29%2F%282%2Api%29 ALIGN=MIDDLE  ALT=\"r+=+%281+-+4x%29%2F%282%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let area of circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A+=+pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+pi%2A%281+-+4x%29%5E2%2F%282%2Api%29%5E2 ALIGN=MIDDLE  ALT=\"A+=+pi%2A%281+-+4x%29%5E2%2F%282%2Api%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28pi%2A%281+-+8x+%2B+16x%5E2%29%29%2F%284%2Api%5E2%29 ALIGN=MIDDLE  ALT=\"A+=+%28pi%2A%281+-+8x+%2B+16x%5E2%29%29%2F%284%2Api%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%2816x%5E2+-+8x+%2B+1%29%2F%284%2Api%29 ALIGN=MIDDLE  ALT=\"A+=+%2816x%5E2+-+8x+%2B+1%29%2F%284%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total area is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%2816x%5E2+-+8x+%2B+1%29%2F%284%2Api%29+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"A+=+%2816x%5E2+-+8x+%2B+1%29%2F%284%2Api%29+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Api ALIGN=MIDDLE  ALT=\"4%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Api%2AA+=+%2816+%2B+4%2Api%29%2Ax%5E2+-+8x+%2B+1 ALIGN=MIDDLE  ALT=\"4%2Api%2AA+=+%2816+%2B+4%2Api%29%2Ax%5E2+-+8x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If I make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Api%2AA ALIGN=MIDDLE  ALT=\"4%2Api%2AA\"   DISABLED_style= \"max-width:90%; height:auto;\" > a minimum, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be a minimum<BR>\n" );
document.write( "With a parabola, the minimum occurs at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bmin%5D+=+-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5Bmin%5D+=+-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=+-8 ALIGN=MIDDLE  ALT=\"b=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+16+%2B+4%2Api ALIGN=MIDDLE  ALT=\"a+=+16+%2B+4%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bmin%5D+=+-%28-8%29%2F%282%2A%2816+%2B+4%2Api%29%29 ALIGN=MIDDLE  ALT=\"x%5Bmin%5D+=+-%28-8%29%2F%282%2A%2816+%2B+4%2Api%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bmin%5D+=+8%2F%2832+%2B+8%2Api%29 ALIGN=MIDDLE  ALT=\"x%5Bmin%5D+=+8%2F%2832+%2B+8%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bmin%5D+=+1%2F%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"x%5Bmin%5D+=+1%2F%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the square is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+4%2F%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"4x+=+4%2F%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the circle is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+%284%2F%284+%2B+pi%29%29+=+%284+%2B+pi%29%2F%284+%2B+pi%29+-+%284%2F%284+%2B+pi%29%29 ALIGN=MIDDLE  ALT=\"1+-+%284%2F%284+%2B+pi%29%29+=+%284+%2B+pi%29%2F%284+%2B+pi%29+-+%284%2F%284+%2B+pi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284+%2B+pi+-+4%29+%2F+%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"%284+%2B+pi+-+4%29+%2F+%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"pi%2F%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cut should be made so the lengths of<BR>\n" );
document.write( "the two wires are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"4%2F%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F%284+%2B+pi%29 ALIGN=MIDDLE  ALT=\"pi%2F%284+%2B+pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Hope I didn't make a mistake- this is the way to do it, though </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );