document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155324>Question 1155324</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A metal box with a square bottom and top is to contain 768 cubic centimeters. The bottom material must be stronger than the rest of the box and costs four cents per square centimeter. Material for the sides and tops is less expensive and costs two cents per square centimeter. Find the dimensions of the box that satisfy the volume requirements at a minimum cost of materials. What is this minimum cost? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777907 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=777907\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A metal box with a square bottom and top is to contain 768 cubic centimeters.<BR>\n" );
document.write( " The bottom material must be stronger than the rest of the box and costs four cents per square centimeter.<BR>\n" );
document.write( " Material for the sides and tops is less expensive and costs two cents per square centimeter.<BR>\n" );
document.write( " Find the dimensions of the box that satisfy the volume requirements at a minimum cost of materials. What is this minimum cost?<BR>\n" );
document.write( ":<BR>\n" );
document.write( "let s = size of the side of the square bottom and square top<BR>\n" );
document.write( "let h = the height of the box<BR>\n" );
document.write( ":<BR>\n" );
document.write( "s^2 * h = the volume<BR>\n" );
document.write( "s^2 * h = 768<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=768%2Fs%5E2 ALIGN=MIDDLE  ALT=\"768%2Fs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ":<BR>\n" );
document.write( "Surface area = bottom area + 4 sides area + top area<BR>\n" );
document.write( "SA = s^2 + 4(hs) + s^2<BR>\n" );
document.write( "replace h<BR>\n" );
document.write( "SA = s^2 + 4(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=768%2Fs%5E2 ALIGN=MIDDLE  ALT=\"768%2Fs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >*s) + s^2<BR>\n" );
document.write( "cancel s<BR>\n" );
document.write( "SA = s^2 + 4(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=768%2Fs ALIGN=MIDDLE  ALT=\"768%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >) + s^2<BR>\n" );
document.write( "SA = s^2 + (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Fs ALIGN=MIDDLE  ALT=\"3072%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >) + s^2<BR>\n" );
document.write( "find the cost<BR>\n" );
document.write( "cost = bottom + 4 sides + top<BR>\n" );
document.write( "Cost = .04(s^2) + .02(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Fs ALIGN=MIDDLE  ALT=\"3072%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >) + .02s^2<BR>\n" );
document.write( "Cost = (.04s^2) + (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=61.44%2Fs ALIGN=MIDDLE  ALT=\"61.44%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >) + .02s^2<BR>\n" );
document.write( "Cost = (.06s^2) + (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=61.44%2Fs ALIGN=MIDDLE  ALT=\"61.44%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "graph this equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+16%2C+-10%2C+30%2C+.06x%5E2%2B%2861.44%2Fx%29%2C+11.52%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+16%2C+-10%2C+30%2C+.06x%5E2%2B%2861.44%2Fx%29%2C+11.52%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Minimum occurs when s = 8 cm<BR>\n" );
document.write( "Find h<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=768%2F8%5E2 ALIGN=MIDDLE  ALT=\"768%2F8%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h = 12 cm<BR>\n" );
document.write( "dimensions then: 8 by 8 by 12 cm<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Find the actual cost<BR>\n" );
document.write( "Cost = .06(8^2) + .02(4(8*12))<BR>\n" );
document.write( "Cost = 3.84 + 7.68<BR>\n" );
document.write( "cost ~ $11.52 is the min cost of the box (green line)\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );