document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=52505>Question 52505</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An open-top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.<BR>\n" );
document.write( "a) 	Find the function V that represents the volume of the box in terms of x.<BR>\n" );
document.write( "Answer	\r<BR>\n" );
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document.write( "b) 	Graph this function and show the graph over the valid range of the variable x..<BR>\n" );
document.write( "Show Graph here\r<BR>\n" );
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document.write( "c) 	Using the graph, what is the value of x that will produce the maximum volume?<BR>\n" );
document.write( "Answer	\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #35064 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=35064\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the volume of such a box, multiply the width by the length by the height.<BR>\n" );
document.write( "The width, after cutting out the corner squares of x by x feet, will be (6-2x) feet and the length will be (8-2x) feet, and, of course, the height of the box will be x feet. So, the volume is expressed by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%288-2x%29%286-2x%29x ALIGN=MIDDLE  ALT=\"V+=+%288-2x%29%286-2x%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand this to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2848-16x-12x%2B4x%5E2%29x ALIGN=MIDDLE  ALT=\"V+=+%2848-16x-12x%2B4x%5E2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplifying this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+4x%5E3-28x%5E2%2B48x ALIGN=MIDDLE  ALT=\"V+=+4x%5E3-28x%5E2%2B48x\"   DISABLED_style= \"max-width:90%; height:auto;\" > This represents the volume of the box expressed in terms of x.\r<BR>\n" );
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document.write( "The graph of this cubic function looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-5%2C6%2C-5%2C25%2C4x%5E3-28x%5E2%2B48x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-5%2C6%2C-5%2C25%2C4x%5E3-28x%5E2%2B48x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The valid range of x is x = 0 to x = 3\r<BR>\n" );
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document.write( "Using the graph to find the value of x that will produce the maximum volume is a little more than x = 1.<BR>\n" );
document.write( "The actual value can be found with a little elementary differential calculus, because it's a matter of finding the \"relative\" maximum of the cubic curve.<BR>\n" );
document.write( "Take the first derivative of the function and set it equal to zero, then solve for x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+12x%5E2-56x%2B48 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+12x%5E2-56x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set this equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-56x%2B48+=+0 ALIGN=MIDDLE  ALT=\"12x%5E2-56x%2B48+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The roots are:<BR>\n" );
document.write( "x = 3.535... Ignore this solution as x is too large.<BR>\n" );
document.write( "x = 1.131... This is the approximate value of x that will produce the largest volume. </SPAN>\n" );
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