document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=371582>Question 371582</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cardboard box has a square base with each edge of the box having length of x inches. The total length of all 12 edges of the box is 144in. Express the volume of the box as a function of x. Express the height in terms of x.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #264761 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=264761\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Added to this solution as requested by student. Please let me know if you got<BR>\n" );
document.write( "this. I am not sure when we edit a solution, that the student gets it.<BR>\n" );
document.write( "ankor@att.net is my address now.<BR>\n" );
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document.write( "A cardboard box has a square base with each edge of the box having length of x inches.<BR>\n" );
document.write( " The total length of all 12 edges of the box is 144in.<BR>\n" );
document.write( " Express the volume of the box as a function of x.<BR>\n" );
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document.write( "Given x = length of the square sides of the base<BR>\n" );
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document.write( "let h = the height of the box<BR>\n" );
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document.write( "then the 12 side total:<BR>\n" );
document.write( "8x + 4h = 144<BR>\n" );
document.write( "simplify, divide by 4<BR>\n" );
document.write( "2x + h = 36<BR>\n" );
document.write( "h = (36-2x)<BR>\n" );
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document.write( "The volume: <BR>\n" );
document.write( "V = x^2*h<BR>\n" );
document.write( "Replace h with (36-2x)<BR>\n" );
document.write( "V = x^2(36-2x)<BR>\n" );
document.write( "V = -2x^3 + 36x^2, (volume in terms of x)<BR>\n" );
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document.write( "Express the height in terms of x. <BR>\n" );
document.write( "Above: h = (36-2x)<BR>\n" );
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document.write( "Response to student comment for length of x for max volume.<BR>\n" );
document.write( "the easy way is to put the vol equation into a graph calc: y = -2x^3 + 36x^2<BR>\n" );
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document.write( "Looks something like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-10%2C+20%2C+-500%2C+2000%2C+-2x%5E3%2B36x%5E2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-10%2C+20%2C+-500%2C+2000%2C+-2x%5E3%2B36x%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can see max area will occur when x=12 inches, Vol will be 1728 cu/inches<BR>\n" );
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document.write( "You can see that max volume occurs when the box is a cube, 12*12*12 = 1728<BR>\n" );
document.write( "(12 edges of 12 inches each) </SPAN>\n" );
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