document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=356869>Question 356869</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An open box with a square base is required to have a volume of 10 cubic feet. Assume the box is to be made from a square piece of cardboard that is has original dimensions (x + 2h)-by-(x + 2h) by cutting out 4 h-by-h squares on the corners of the cardboard and folding up the sides where h is the height of the open box sides (see figure below)\r<BR>\n" );
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document.write( "Base length x                                                                                 h by h square cut out to form open box<BR>\n" );
document.write( "                                                                                               <BR>\n" );
document.write( "A] Express the surface area A(x) of the box as a function of the length of the base x.\r<BR>\n" );
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document.write( "B] Use a graphing utility to graph A(x) and determine the dimensions of an open box with the smallest surface area possible. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #255121 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=255121\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An open box with a square base is required to have a volume of 10 cubic feet.<BR>\n" );
document.write( " Assume the box is to be made from a square piece of cardboard that is has<BR>\n" );
document.write( " original dimensions (x + 2h)-by-(x + 2h)<BR>\n" );
document.write( " by cutting out 4 h-by-h squares on the corners of the cardboard and folding<BR>\n" );
document.write( " up the sides where h is the height of the open box sides <BR>\n" );
document.write( "Base length x h by h square cut out to form open box<BR>\n" );
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document.write( "The removal of the h by h corners, reduces the dimension by 2h, therefore<BR>\n" );
document.write( "The box dimensions will x by x by h, therefore:<BR>\n" );
document.write( "Vol = x^2*h<BR>\n" );
document.write( "Given the vol as 10 cu/ft:<BR>\n" );
document.write( "x^2*h = 10 cu/ft<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Fx%5E2 ALIGN=MIDDLE  ALT=\"10%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A] Express the surface area A(x) of the box as a function of the length of the base x.<BR>\n" );
document.write( "Surface area of a 5 sided (open) box:<BR>\n" );
document.write( "S.A. = x^2 + 4(x*h)<BR>\n" );
document.write( "replace h with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Fx%5E2 ALIGN=MIDDLE  ALT=\"10%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "S.A. = x^2 + 4(x*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Fx%5E2 ALIGN=MIDDLE  ALT=\"10%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "Cancel x<BR>\n" );
document.write( "S.A. = x^2 + 4(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Fx ALIGN=MIDDLE  ALT=\"10%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "S.A = x^2 + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%2Fx ALIGN=MIDDLE  ALT=\"40%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >; the surface area as a function of x<BR>\n" );
document.write( ":<BR>\n" );
document.write( "B] Use a graphing utility to graph A(x) and determine the dimensions of an open box with the smallest surface area possible.<BR>\n" );
document.write( "In a TI83 or similar enter y= x^2+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%2Fx ALIGN=MIDDLE  ALT=\"40%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I used a scale: -2,+10; -10,+50<BR>\n" );
document.write( "Looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-2%2C+10%2C+-10%2C+50%2C+x%5E2%2B%2840%2Fx%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-2%2C+10%2C+-10%2C+50%2C+x%5E2%2B%2840%2Fx%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Using the min feature on the calc, I got x=2.744, surface area of 22.1 is min<BR>\n" );
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document.write( "Using x=2.744,<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F2.744%5E2 ALIGN=MIDDLE  ALT=\"10%2F2.744%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h ~ 1.33 is the height<BR>\n" );
document.write( "Find the Surface area from these values<BR>\n" );
document.write( "S.A. = 2.744^2 + 4(2.744*1.334)<BR>\n" );
document.write( "S.A. = 7.53 + 4(3.65)<BR>\n" );
document.write( "S.A. = 7.53 + 14.6<BR>\n" );
document.write( "S.A. = 22.13 which agrees with the Calc values<BR>\n" );
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document.write( "Did all this make sense to you? </SPAN>\n" );
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