document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934802>Question 934802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A  right triangle has a fixed hypotenuse of 30 cm. And the other two sides are allowed to vary. Determine the largest possible area of the triangle.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #567838 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=567838\">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=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of one leg, in cm.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of the other leg, in cm.<BR>\n" );
document.write( "}}}According to the Pythagorean theorem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=30%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=30%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=900 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=900\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=900-x%5E2 ALIGN=MIDDLE  ALT=\"y%5E2=900-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the right triangle, in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2Ax%2Ay ALIGN=MIDDLE  ALT=\"area=%281%2F2%29%2Ax%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%28%281%2F2%29%2Ax%2Ay%29%5E2 ALIGN=MIDDLE  ALT=\"area%5E2=%28%281%2F2%29%2Ax%2Ay%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=area%5E2=%281%2F2%29%5E2%2Ax%5E2%2Ay%2A2 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F2%29%5E2%2Ax%5E2%2Ay%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%2Ax%5E2%2Ay%2A2 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%2Ax%5E2%2Ay%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900-x%5E2 ALIGN=MIDDLE  ALT=\"900-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (found above) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%2Ax%5E2%2A%28900-x%5E2%29 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%2Ax%5E2%2A%28900-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%28-x%5E4%2B900x%5E2%29 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%28-x%5E4%2B900x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%28-%28x%5E2%29%5E2%2B900x%5E2%29 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%28-%28x%5E2%29%5E2%2B900x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That last expression shows <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2 ALIGN=MIDDLE  ALT=\"area%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > a as quadratic function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and we know that it will have a maximum when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=900%2F2 ALIGN=MIDDLE  ALT=\"x%5E2=900%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=450 ALIGN=MIDDLE  ALT=\"x%5E2=450\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For that value of <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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%28-450%5E2%2B900%2A450%29 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%28-450%5E2%2B900%2A450%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%2A450%2A%28-450%2B900%29 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%2A450%2A%28-450%2B900%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%2A450%2A450 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%2A450%2A450\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%281%2F4%29%2A450%5E2 ALIGN=MIDDLE  ALT=\"area%5E2=%281%2F4%29%2A450%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=450%5E2%2F2%5E2 ALIGN=MIDDLE  ALT=\"area%5E2=450%5E2%2F2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5E2=%28450%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"area%5E2=%28450%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=area%5E2=225%5E2 ALIGN=MIDDLE  ALT=\"area%5E2=225%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28area=225%29 ALIGN=MIDDLE  ALT=\"highlight%28area=225%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that is the largest possible area of the triangle, in square centimeters. </SPAN>\n" );
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