document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=237557>Question 237557</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a rectangular sheet of cardboard that is 15 inches by 25 inches, a small square of the same size is cut from each corner.  Each side of the cardboard is folded along the cuts to form a lidless box.\r<BR>\n" );
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document.write( "A)What is the maximum volume, V(x), of the box?\r<BR>\n" );
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document.write( "B)What size of the cut squares would produce a box with a volume equal to 400 cubic inches? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174706 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=174706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Answer to B:\r<BR>\n" );
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document.write( "None.  \r<BR>\n" );
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document.write( "Maximum Volume of the box is going to be less than 600 cubic inches.\r<BR>\n" );
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document.write( "The maximum volume of the box is going to be somewhere around 100.\r<BR>\n" );
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document.write( "Here's how I figured.\r<BR>\n" );
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document.write( "Let x = the length of one side of each square.\r<BR>\n" );
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document.write( "If you cut a square off each corner of the rectangle, then the length will be reduced by 2x and the width will be reduced by 2x.\r<BR>\n" );
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document.write( "The height of your box will be x.\r<BR>\n" );
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document.write( "the length of your box will be 25 - 2x.\r<BR>\n" );
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document.write( "The width of your box will be 15 - 2x.\r<BR>\n" );
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document.write( "The volume of your box is calculated as V = (25-2x) * (15-2x) * x\r<BR>\n" );
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document.write( "To find the rots of this equation, set it equal to 0 and solve.\r<BR>\n" );
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document.write( "You get:\r<BR>\n" );
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document.write( "(-2x+25) * (-2x+15) * x = 0\r<BR>\n" );
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document.write( "This equation would cross the x-axis at:\r<BR>\n" );
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document.write( "x = 0 and x = 12.5 and x = 7.5\r<BR>\n" );
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document.write( "A graph of this equation would look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-20%2C20%2C-600%2C600%2C%28-2x%2B25%29%2A%28-2x%2B15%29%2Ax%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-20%2C20%2C-600%2C600%2C%28-2x%2B25%29%2A%28-2x%2B15%29%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The domain of this equation has to be x > 0 to x < 7.5.\r<BR>\n" );
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document.write( "When x = 0, you have a flat sheet of cardboard with no height.\r<BR>\n" );
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document.write( "When x = 7.5, the width would be equal to 0 because 15 - 15 = 0.\r<BR>\n" );
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document.write( "This means that the range of this equation would be a minimum height > 0 and a maximum height < 7.5\r<BR>\n" );
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document.write( "The volume of the box would have to be be a minimum > 0 and a maximum of around 500 cubic inches as shown on the graph.\r<BR>\n" );
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document.write( "This would occur at somewhere around x = 3\r<BR>\n" );
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document.write( "Without looking at the graph, this would be difficult to solve.\r<BR>\n" );
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document.write( "It was only after looking at the graph that I was able to see what was going on.\r<BR>\n" );
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document.write( "There is a maximum point but I don't know the formula to find it given the equation as it stands.\r<BR>\n" );
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document.write( "It's not a quadratic and doesn't look like it can be made into a quadratic easily.\r<BR>\n" );
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document.write( "Best I can do.\r<BR>\n" );
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document.write( "Hope it helps just a little.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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