document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1058420>Question 1058420</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A one compartment vertical file is to be constructed by bending the long side of an 8 in by 12 in. sheet of plastic along two lines to form a U shape. How tall should the file be to maximize the volume that it can&#8203; hold? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #673506 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=673506\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The cross section would be a squared up U, with vertical sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches tall,<BR>\n" );
document.write( "and a middle horizontal section <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-2x ALIGN=MIDDLE  ALT=\"12-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches wide.<BR>\n" );
document.write( "We really want to maximize the cross section area of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-2x ALIGN=MIDDLE  ALT=\"12-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > rectangle,<BR>\n" );
document.write( "because the 8 inch depth of the file is determined by the width of the 8 in by 12 in sheet of plastic.<BR>\n" );
document.write( "The area of that cross section is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=2x%2812-2x%29 ALIGN=MIDDLE  ALT=\"A%28x%29=2x%2812-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=4x%286-x%29 ALIGN=MIDDLE  ALT=\"A%28x%29=4x%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is a polynomial of degree 2, or in other words a quadratic function.<BR>\n" );
document.write( "We know that those are symmetrical functions with a minimum or maximum at their axis of symmetry (at the vertex of the parabola).<BR>\n" );
document.write( "That maximum has to be exactly halfway between the zeros of the function,<BR>\n" );
document.write( "which we knew would be at <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;\" > and (((x=6}}} before writing anything,<BR>\n" );
document.write( "because those values correspond to no vertical sides, and no horizontal bottom respectively.<BR>\n" );
document.write( "So, the file should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%280%2B6%29%2F2=highlight%283%29 ALIGN=MIDDLE  ALT=\"x=%280%2B6%29%2F2=highlight%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches tall.<BR>\n" );
document.write( " <BR>\n" );
document.write( "What did your teacher expect?<BR>\n" );
document.write( "Maybe calculating volume as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=8%2Ax%2A%2812-2x%29 ALIGN=MIDDLE  ALT=\"V%28x%29=8%2Ax%2A%2812-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=8%2A%2812x-2x%5E2%29 ALIGN=MIDDLE  ALT=\"V%28x%29=8%2A%2812x-2x%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=V%28x%29=-16x%5E2%2B96x ALIGN=MIDDLE  ALT=\"V%28x%29=-16x%5E2%2B96x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and after that,<BR>\n" );
document.write( "either calculating the derivative to find the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > value for the maximum,<BR>\n" );
document.write( "or using the \"formula\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to find the axis of symmetry of a quadratic function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );