document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194553>Question 194553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A wire 20 feet long is to be cut into two pieces. one pierce will be shaped as a square and the other piece will be shaped as an equilateral triangle. express the total area A enclosed by by the piece of wire as a function of the length x of a side of the equilateral triangle. what is the domain of A? for what value of x will the total area A be the smallest?<BR>\n" );
document.write( "please i need help. please help me please.....  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #146040 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=146040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a side of the equilateral triangle, <BR>\n" );
document.write( "then the length of wire used for the triangle is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
document.write( "The length of wire left over for the square is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+-+3x ALIGN=MIDDLE  ALT=\"20+-+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
document.write( "One side of that square would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820+-+3x%29%2F4 ALIGN=MIDDLE  ALT=\"%2820+-+3x%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
document.write( "If <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 the total area, then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%283%29%2A%28x%2F2%29%5E2+%2B+%28%2820+-+3x%29%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"A+=+sqrt%283%29%2A%28x%2F2%29%5E2+%2B+%28%2820+-+3x%29%2F4%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+=+sqrt%283%29%2A%28x%2F2%29%5E2+%2B+%28%2820+-+3x%29%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"A+=+sqrt%283%29%2A%28x%2F2%29%5E2+%2B+%28%2820+-+3x%29%2F4%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+=+%28%28sqrt%283%29%29%2F4%29%2Ax%5E2+%2B+%28400+-+120x+%2B+9x%5E2%29+%2F+16 ALIGN=MIDDLE  ALT=\"A+=+%28%28sqrt%283%29%29%2F4%29%2Ax%5E2+%2B+%28400+-+120x+%2B+9x%5E2%29+%2F+16\"   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=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16A+=+4%2Asqrt%283%29%2Ax%5E2+%2B+%28400+-+120x+%2B+9x%5E2%29%2F16 ALIGN=MIDDLE  ALT=\"16A+=+4%2Asqrt%283%29%2Ax%5E2+%2B+%28400+-+120x+%2B+9x%5E2%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28sqrt%283%29%2F4+%2B+9%29%2Ax%5E2+-+120x+%2B+400 ALIGN=MIDDLE  ALT=\"A+=+%28sqrt%283%29%2F4+%2B+9%29%2Ax%5E2+-+120x+%2B+400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------------------------------<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or I had no triangle at all, then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+400 ALIGN=MIDDLE  ALT=\"A+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as it should, since the whole <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
document.write( "is being used by the squareand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+20%5E2 ALIGN=MIDDLE  ALT=\"A+=+20%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20%2F3 ALIGN=MIDDLE  ALT=\"x+=+20%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then nothing is left for the square, and<BR>\n" );
document.write( "all the area is used by the triangle.<BR>\n" );
document.write( "I'll check this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%283%29%2A%28x%2F2%29%5E2+ ALIGN=MIDDLE  ALT=\"A+=+sqrt%283%29%2A%28x%2F2%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+=+1.732%2A%2820%2F6%29%5E2 ALIGN=MIDDLE  ALT=\"A+=+1.732%2A%2820%2F6%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+=+1.732%2A11.111 ALIGN=MIDDLE  ALT=\"A+=+1.732%2A11.111\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+19.244 ALIGN=MIDDLE  ALT=\"A+=+19.244\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft2<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28sqrt%283%29%2F4+%2B+9%29%2Ax%5E2+-+120x+%2B+400 ALIGN=MIDDLE  ALT=\"A+=+%28sqrt%283%29%2F4+%2B+9%29%2Ax%5E2+-+120x+%2B+400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28sqrt%283%29%2F4+%2B+9%29%2A%2820%2F3%29%5E2+-+120%2A%2820%2F3%29+%2B+400 ALIGN=MIDDLE  ALT=\"A+=+%28sqrt%283%29%2F4+%2B+9%29%2A%2820%2F3%29%5E2+-+120%2A%2820%2F3%29+%2B+400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28sqrt%283%29%2F4+%2B+9%29%2A%28400%2F9%29+-+800+%2B+400 ALIGN=MIDDLE  ALT=\"A+=+%28sqrt%283%29%2F4+%2B+9%29%2A%28400%2F9%29+-+800+%2B+400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+100%2Asqrt%283%29%2F9+%2B+400+-+800+%2B+400+ ALIGN=MIDDLE  ALT=\"A+=+100%2Asqrt%283%29%2F9+%2B+400+-+800+%2B+400+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+19.244 ALIGN=MIDDLE  ALT=\"A+=+19.244\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft2<BR>\n" );
document.write( "The domain of <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 19.244 < A < 400<BR>\n" );
document.write( "--------------------------------------------<BR>\n" );
document.write( "When an equation is of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the minimum is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%28b%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"x+=+-%28b%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-120 ALIGN=MIDDLE  ALT=\"b+=+-120\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+sqrt%283%29%2F4+%2B+9 ALIGN=MIDDLE  ALT=\"a+=+sqrt%283%29%2F4+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28b%2F%282a%29%29+=+-%28-120%29%2F%282%2A%28sqrt%283%29%29%2F4+%2B+18%29 ALIGN=MIDDLE  ALT=\"-%28b%2F%282a%29%29+=+-%28-120%29%2F%282%2A%28sqrt%283%29%29%2F4+%2B+18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28b%2F%282a%29%29+=+120+%2F+18.866 ALIGN=MIDDLE  ALT=\"-%28b%2F%282a%29%29+=+120+%2F+18.866\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28b%2F%282a%29%29+=+6.361 ALIGN=MIDDLE  ALT=\"-%28b%2F%282a%29%29+=+6.361\"   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
document.write( "This is the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for which <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 minimum<BR>\n" );
document.write( "I'll graph the function to check this (approximately)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-5%2C+16%2C+-15%2C+450%2C+9.433x%5E2+-+120x+%2B+400%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-5%2C+16%2C+-15%2C+450%2C+9.433x%5E2+-+120x+%2B+400%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "And here's a close-up<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-2%2C+9%2C+-5%2C+25%2C+9.433x%5E2+-+120x+%2B+400%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-2%2C+9%2C+-5%2C+25%2C+9.433x%5E2+-+120x+%2B+400%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );