document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1072946>Question 1072946</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle has two equal sides and. Third side. Determine the ratio between the sides a and b to enclose the maximum area for a given total length of the sides? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #687803 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=687803\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Look at an isosceles triangle with sides, m, m, and 2n.<BR>\n" );
document.write( "So that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=a ALIGN=MIDDLE  ALT=\"m=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=b%2F2 ALIGN=MIDDLE  ALT=\"n=b%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter is then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=m%2Bm%2B2n ALIGN=MIDDLE  ALT=\"P=m%2Bm%2B2n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=2m%2B2n ALIGN=MIDDLE  ALT=\"P=2m%2B2n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter is set to a constant length, 2L.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2m%2B2n=2L ALIGN=MIDDLE  ALT=\"2m%2B2n=2L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bn=L ALIGN=MIDDLE  ALT=\"m%2Bn=L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=L-m ALIGN=MIDDLE  ALT=\"n=L-m\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area would then be,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29%28Base%29%28Height%29 ALIGN=MIDDLE  ALT=\"A=%281%2F2%29%28Base%29%28Height%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=%281%2F2%29n%28sqrt%28m%5E2-n%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=%281%2F2%29n%28sqrt%28m%5E2-n%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting for n,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29%28L-m%29%28sqrt%28m%5E2-%28L-m%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=%281%2F2%29%28L-m%29%28sqrt%28m%5E2-%28L-m%29%5E2%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=A=%281%2F2%29%28L-m%29%28sqrt%282mL-L%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=%281%2F2%29%28L-m%29%28sqrt%282mL-L%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So to find the maximum area, take the derivative of A with respect to a and set it equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fda=%28L%282L-3m%29%29%2F%282%28sqrt%282mL-L%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"dA%2Fda=%28L%282L-3m%29%29%2F%282%28sqrt%282mL-L%5E2%29%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=%28L%282L-3m%29%29=0 ALIGN=MIDDLE  ALT=\"%28L%282L-3m%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "One solution,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2L-3m=0 ALIGN=MIDDLE  ALT=\"2L-3m=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3m=2L ALIGN=MIDDLE  ALT=\"3m=2L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%282%2F3%29L ALIGN=MIDDLE  ALT=\"m=%282%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=L-%282%2F3%29L ALIGN=MIDDLE  ALT=\"n=L-%282%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%281%2F3%29L ALIGN=MIDDLE  ALT=\"n=%281%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So in terms of a and b, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=a ALIGN=MIDDLE  ALT=\"m=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=b%2F2 ALIGN=MIDDLE  ALT=\"n=b%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So substituting,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%282%2F3%29L ALIGN=MIDDLE  ALT=\"a=%282%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F2=%281%2F3%29L ALIGN=MIDDLE  ALT=\"b%2F2=%281%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=%282%2F3%29L ALIGN=MIDDLE  ALT=\"b=%282%2F3%29L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the ratio is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb=%28%282%2F3%29L%29%2F%28%282%2F3%29L%29 ALIGN=MIDDLE  ALT=\"a%2Fb=%28%282%2F3%29L%29%2F%28%282%2F3%29L%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a%2Fb=1%29 ALIGN=MIDDLE  ALT=\"highlight%28a%2Fb=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );