document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119021>Question 119021</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the base and height of a triangle is 25 cm. What is the maximum area of the triangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87133 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have two equations:\r<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bh=25 ALIGN=MIDDLE  ALT=\"b%2Bh=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=bh%2F2 ALIGN=MIDDLE  ALT=\"A=bh%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And we want to maximize A.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From 1) we can say:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=25-b ALIGN=MIDDLE  ALT=\"h=25-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and with this information we can re-write 2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=b%2825-b%29=25b-b%5E2 ALIGN=MIDDLE  ALT=\"A=b%2825-b%29=25b-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-b%5E2%2B25b%2B0 ALIGN=MIDDLE  ALT=\"A=-b%5E2%2B25b%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > describes a concave down (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) parabola with vertex at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-25%2F%282%2A%28-1%29%29=12.5 ALIGN=MIDDLE  ALT=\"b=-25%2F%282%2A%28-1%29%29=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since this is a concave down parabola, we know that the vertex point is a maximum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(Alternatively, if you are taking calculus at this point, you can set the first derivative equal to zero and solve for x to find the x coordinate of the local minimum/maximum, and then evaluate the sign of the second derivative to determine that the point is a maximum, thus:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdb=25-2b ALIGN=MIDDLE  ALT=\"dA%2Fdb=25-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25-2b=0 ALIGN=MIDDLE  ALT=\"25-2b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12.5 ALIGN=MIDDLE  ALT=\"b=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2%28A%29%2Fdb%5E2=-2%3C0 ALIGN=MIDDLE  ALT=\"d%5E2%28A%29%2Fdb%5E2=-2%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the point is a maximum)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have determined the length of the base that produces the maximum volume, so we can calculate the height by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=25-12.5=12.5 ALIGN=MIDDLE  ALT=\"h=25-12.5=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  And finally, the maximum Area by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=%2812.5%5E2%29%2F2=156.25%2F2=78.125cm%5E2 ALIGN=MIDDLE  ALT=\"A%5Bmax%5D=%2812.5%5E2%29%2F2=156.25%2F2=78.125cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You should actually state your answer as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=78cm%5E2 ALIGN=MIDDLE  ALT=\"78cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > because your answer should never be stated with more precision than the least precise measurement given in the problem.  Since the sum of the base and height were given to the nearest whole centimeter, your answer should also be to the nearest whole centimeter. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );