document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151928>Question 1151928</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a box with a square base has no top. if 64 cm squared material is used, what is the maximum possible volume for the box? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773862 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=773862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x be the side of the square base, and let y be the height (depth) of the box.<br><BR>\n" );
document.write( "We are to maximize the volume, which is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28x%5E2%29%28y%29 ALIGN=MIDDLE  ALT=\"V+=+%28x%5E2%29%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We need that volume formula to be in a single variable.  Use the given information about the amount of material used to make the box (bottom and four sides) to solve for y in terms of x and substitute in the volume formula.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4xy+=+64 ALIGN=MIDDLE  ALT=\"x%5E2%2B4xy+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the total area of the base and four sides is 64 square cm]<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4xy+=+64-x%5E2 ALIGN=MIDDLE  ALT=\"4xy+=+64-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=y+=+%2864-x%5E2%29%2F%284x%29+=+16%2Fx-x%2F4 ALIGN=MIDDLE  ALT=\"y+=+%2864-x%5E2%29%2F%284x%29+=+16%2Fx-x%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28x%5E2%29%28y%29+=+%28x%5E2%29%2816%2Fx-x%2F4%29+=+16x-x%5E3%2F4 ALIGN=MIDDLE  ALT=\"V+=+%28x%5E2%29%28y%29+=+%28x%5E2%29%2816%2Fx-x%2F4%29+=+16x-x%5E3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Find the derivative and set it to zero to find the value of x that maximizes the volume.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+16-3x%5E2%2F4 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+16-3x%5E2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-3x%5E2%2F4+=+0 ALIGN=MIDDLE  ALT=\"16-3x%5E2%2F4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2F4+=+16 ALIGN=MIDDLE  ALT=\"3x%5E2%2F4+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+=+64 ALIGN=MIDDLE  ALT=\"3x%5E2+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+64%2F3 ALIGN=MIDDLE  ALT=\"x%5E2+=+64%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+8%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+8%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Determine y for that value of x:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+16%2Fx-x%2F4+=+16%2F%288%2Fsqrt%283%29%29-%288%2Fsqrt%283%29%29%2F4+=+2%2Asqrt%283%29-2%2Fsqrt%283%29+=+2%2Asqrt%283%29-%282%2F3%29sqrt%283%29+=+%284%2F3%29sqrt%283%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Determine the maximum volume:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28x%5E2%29%28y%29+=+%2864%2F3%29%28%284%2F3%29sqrt%283%29%29+=+%28256%2F9%29sqrt%283%29 ALIGN=MIDDLE  ALT=\"V+=+%28x%5E2%29%28y%29+=+%2864%2F3%29%28%284%2F3%29sqrt%283%29%29+=+%28256%2F9%29sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );