document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=633011>Question 633011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a piece of cardboard, twice as long as it is wide. I will cut 4x4 suqares out of each corners, so I fold up the corners and use it as a tray. The volume of the tray will be 96 cubic inches. How large is the piece of cardboard? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #398598 by </B><A HREF=/tutors/aboutme.mpl?userid=graphmatics><B>graphmatics(170)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=graphmatics><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=graphmatics><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=398598\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x be the width of the cardboard. 2x is the length of the card. After cutting out the 4 by 4s on the corners and creating the tray the area of the bottom of the tray will be (x-8)(2x-8). The height of the tray is 4. The volume of the tray is (x-8)(2x-8)4. Because the volume of the tray is 96 we have that (x-8)(2x-8)(4)=96 so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Ax%5E2-24x%2B64%29%284%29=96 ALIGN=MIDDLE  ALT=\"%282%2Ax%5E2-24x%2B64%29%284%29=96\"   DISABLED_style= \"max-width:90%; height:auto;\" >.    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2-24x%2B64=24 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2-24x%2B64=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-12x%2B32=12 ALIGN=MIDDLE  ALT=\"x%5E2-12x%2B32=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.      <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-12x%2B20=0 ALIGN=MIDDLE  ALT=\"x%5E2-12x%2B20=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We solve this quadratic equation. <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-12x%2B20+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-12x%2B20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-12%29%5E2-4%2A1%2A20=64 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-12%29%5E2-4%2A1%2A20=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=64 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--12%2B-sqrt%28+64+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--12%2B-sqrt%28+64+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+10 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-12%29-sqrt%28+64+%29%29%2F2%5C1+=+2 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-12%29-sqrt%28+64+%29%29%2F2%5C1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-12x%2B20 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-12x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-12x%2B20+=+1%28x-10%29%2A%28x-2%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-12x%2B20+=+1%28x-10%29%2A%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10, 2.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-12%2Ax%2B20+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-12%2Ax%2B20+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( ".   We must select x=10 because x must be greater than 8. The cardboard dimensions are 10 and 20. </SPAN>\n" );
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