document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=428043>Question 428043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the least amount of material needed to make a square based open box that has a volume of 5000 cubic meters. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #297599 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=297599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The box is open, so it will have 5 sides.<BR>\n" );
document.write( "Let a = the side length of the square bottom<BR>\n" );
document.write( "Let h = the height of the box<BR>\n" );
document.write( "We are given the volume of the box = 5000.<BR>\n" );
document.write( "We need to find a formula for the amount of material required for the box.<BR>\n" );
document.write( "The bottom will have area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each side will have area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ah ALIGN=MIDDLE  ALT=\"ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and there are 4 of them.<BR>\n" );
document.write( "Thus, the total amount of material will be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M+=+a%5E2+%2B+4ah ALIGN=MIDDLE  ALT=\"M+=+a%5E2+%2B+4ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1)<BR>\n" );
document.write( "The volume of the box, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2h+=+5000 ALIGN=MIDDLE  ALT=\"a%5E2h+=+5000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+5000%2Fa%5E2 ALIGN=MIDDLE  ALT=\"h+=+5000%2Fa%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute this value for h into equation (1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M+=+a%5E2+%2B+4a%285000%2Fa%5E2%29+=+a%5E2+%2B+20000%2Fa ALIGN=MIDDLE  ALT=\"M+=+a%5E2+%2B+4a%285000%2Fa%5E2%29+=+a%5E2+%2B+20000%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To minimize M, we take the derivative and set = 0:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+2a+-+20000%2Fa%5E2 ALIGN=MIDDLE  ALT=\"0+=+2a+-+20000%2Fa%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for a:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3+=+10000+-%3E+a+=+21.544 ALIGN=MIDDLE  ALT=\"a%5E3+=+10000+-%3E+a+=+21.544\"   DISABLED_style= \"max-width:90%; height:auto;\" > m<BR>\n" );
document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+5000%2F21.544%5E2+=+10.772 ALIGN=MIDDLE  ALT=\"h+=+5000%2F21.544%5E2+=+10.772\"   DISABLED_style= \"max-width:90%; height:auto;\" > m\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );