document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=913117>Question 913117</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm not sure where to begin on this. I'm in Algebra II, but I feel like this is more of a pre-calc question. Any ideas? United Parcel Service has contracted you to design a closed box with a square base that<BR>\n" );
document.write( "has a volume of 10,000 cubic inches. Initially they would like to know the dimensions of such a<BR>\n" );
document.write( "box with minimum surface area. They are considering creating the box with a reinforced bottom.<BR>\n" );
document.write( "The box would be made out of a cardboard material costing 1cent per square inch to build the walls<BR>\n" );
document.write( "and top, however the bottom would be constructed with stronger material, costing 2cents per square<BR>\n" );
document.write( "inch to build. How do the dimensions of the box having the minimum surface area compare to the<BR>\n" );
document.write( "dimensions of the reinforced box which costs the least? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #554318 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39616)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=554318\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The beginning for both of the functions first attends volume.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Bottom is a square shape, so if x is the sidelength for the bottom (as well as top), and if y is height of box, then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000=yx%5E2 ALIGN=MIDDLE  ALT=\"10000=yx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and from it, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10000%2Fx%5E2 ALIGN=MIDDLE  ALT=\"y=10000%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "TOTAL SURFACE AREA\r<BR>\n" );
document.write( "\n" );
document.write( "Let A be variable for total surface area, to be formed into a function of x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2x%5E2%2B4xy ALIGN=MIDDLE  ALT=\"A=2x%5E2%2B4xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2x%5E2%2B4x%2810000%2Fx%5E2%29 ALIGN=MIDDLE  ALT=\"A=2x%5E2%2B4x%2810000%2Fx%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=highlight_green%28A=2x%5E2%2B40000%2Fx%29 ALIGN=MIDDLE  ALT=\"highlight_green%28A=2x%5E2%2B40000%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which you might possibly want as a single numerator expression and a single denominator expression.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "COST FUNCTION\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1 cent per sq in. for walls and top; 2 cents per sq. in. for bottom.<BR>\n" );
document.write( "Let C be the cost for the box.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=%28x%5E2%29%282%29%2B%28x%5E2%29%281%29%2B4xy%2A1 ALIGN=MIDDLE  ALT=\"C=%28x%5E2%29%282%29%2B%28x%5E2%29%281%29%2B4xy%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Bx%5E2%2B4xy ALIGN=MIDDLE  ALT=\"2x%5E2%2Bx%5E2%2B4xy\"   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%2B4x%2810000%2Fx%5E2%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B4x%2810000%2Fx%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=highlight_green%28C=3x%5E2%2B40000%2Fx%29 ALIGN=MIDDLE  ALT=\"highlight_green%28C=3x%5E2%2B40000%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The form of the area function and the cost function are the same.  Both are rational functions.  They show the same format.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The question to minimize and then compare the dimensions of these two functions seems outside of and beyond Algebra II; but might be used in College Algebra, but more likely fit in Calculus 1.  Very likely most Algebra II students are allowed, encourage, even TAUGHT TO USE a graphing calculator, so if this is something you know how to use, you can find useful results.  Decide on the x value or pick what seems to correspond to each minimum, and use the value to evaluate y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "...<BR>\n" );
document.write( "AREA<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28350%2C350%2C-10%2C35%2C-150%2C5000%2C2x%5E2%2B40000%2Fx%29 ALIGN=MIDDLE  ALT=\"graph%28350%2C350%2C-10%2C35%2C-150%2C5000%2C2x%5E2%2B40000%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "COST<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28350%2C350%2C-10%2C35%2C-150%2C5000%2C3x%5E2%2B40000%2Fx%29 ALIGN=MIDDLE  ALT=\"graph%28350%2C350%2C-10%2C35%2C-150%2C5000%2C3x%5E2%2B40000%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );