document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=242992>Question 242992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular piece of cardboard 11 in by 14 in is made into a box by cutting identical squares from each corner and folding up the sides. If the bottom of the box turns out to have an area of 80 sq. in. what size squares were cut from the corners? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178006 by </B><A HREF=/tutors/aboutme.mpl?userid=JimboP1977><B>JimboP1977(311)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=JimboP1977><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=JimboP1977><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=178006\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The best way to do this is to draw a diagram and mark in dashed lines the corner sections and centre section.\r<BR>\n" );
document.write( "\n" );
document.write( "The whole area of the cardboard is equal to 154 inches^2. The corner sections are equal to 4x^2. The mid sections are equal to 2x(14-2x) and 2x(11-2x).\r<BR>\n" );
document.write( "\n" );
document.write( "So we know that the total area of the cardboard minus the 4 corners minus the 4 mid sections must equal the centre bottom area of 80 inches^2.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=154-4x%5E2-2x%2814-2x%29-2x%2811-2x%29+=+80 ALIGN=MIDDLE  ALT=\"154-4x%5E2-2x%2814-2x%29-2x%2811-2x%29+=+80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=154-4x%5E2-%2828x-4x%5E2%29-%2822x-4x%5E2%29+=+80 ALIGN=MIDDLE  ALT=\"154-4x%5E2-%2828x-4x%5E2%29-%2822x-4x%5E2%29+=+80\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-50x%2B74+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2-50x%2B74+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange to equal zero.\r<BR>\n" );
document.write( "\n" );
document.write( "Use the quadratic formula to find x.\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=4x%5E2%2B-50x%2B74+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-50x%2B74+=+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=%28-50%29%5E2-4%2A4%2A74=1316 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-50%29%5E2-4%2A4%2A74=1316\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1316 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--50%2B-sqrt%28+1316+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--50%2B-sqrt%28+1316+%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-%28-50%29%2Bsqrt%28+1316+%29%29%2F2%5C4+=+10.7845892868043 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-50%29%2Bsqrt%28+1316+%29%29%2F2%5C4+=+10.7845892868043\"   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-%28-50%29-sqrt%28+1316+%29%29%2F2%5C4+=+1.71541071319574 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-50%29-sqrt%28+1316+%29%29%2F2%5C4+=+1.71541071319574\"   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=4x%5E2%2B-50x%2B74 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-50x%2B74\"   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=4x%5E2%2B-50x%2B74+=+4%28x-10.7845892868043%29%2A%28x-1.71541071319574%29 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-50x%2B74+=+4%28x-10.7845892868043%29%2A%28x-1.71541071319574%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10.7845892868043, 1.71541071319574.\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+4%2Ax%5E2%2B-50%2Ax%2B74+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-50%2Ax%2B74+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x cannot be 10.784 as this would mean it was longer than the sides which of course does not make sense! So it must be 1.7154 inches. So the size of the squares cut was 1.7154^2 = 2.942 inches^2. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );