document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=298862>Question 298862</A>: <I><SPAN STYLE=\"word-break: break-all;\"> By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. <BR>\n" );
document.write( "(a) If the cardboard is 15 in. long and 7 in. wide, find the dimensions of the box that will yield the maximum volume.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #214945 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=214945\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First write the equation for the volume of the box.<BR>\n" );
document.write( "If you cut squares from each corner where the side of the square is x inches, then the volume of the resulting box when the flaps are folded up can be expressed as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2815-2x%29%287-2x%29X ALIGN=MIDDLE  ALT=\"V+=+%2815-2x%29%287-2x%29X\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+4x%5E3-44x%5E2%2B105x ALIGN=MIDDLE  ALT=\"V+=+4x%5E3-44x%5E2%2B105x\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the value of x which yields the maximum of volume (V), take the first derivative of this equation and set it equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+12x%5E2-88x%2B105 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+12x%5E2-88x%2B105\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set this equal to zero and solve for x;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-88x%2B105+=+0 ALIGN=MIDDLE  ALT=\"12x%5E2-88x%2B105+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve by the quadratic formula:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a = 12, b = -88, and c = 105. You can do this and you should get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5.833 ALIGN=MIDDLE  ALT=\"x+=+5.833\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.5 ALIGN=MIDDLE  ALT=\"x+=+1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Discard <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5.833 ALIGN=MIDDLE  ALT=\"x+=+5.833\"   DISABLED_style= \"max-width:90%; height:auto;\" > because it would result in a negative value for the length of the bottom of the box: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-2x+=+7-11.66 ALIGN=MIDDLE  ALT=\"7-2x+=+7-11.66\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4.66 ALIGN=MIDDLE  ALT=\"-4.66\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.5 ALIGN=MIDDLE  ALT=\"x+=+1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >inches.<BR>\n" );
document.write( "The dimensions of the box are:<BR>\n" );
document.write( "Length, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+15-2x ALIGN=MIDDLE  ALT=\"L+=+15-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-2%281.5%29+=+15-3 ALIGN=MIDDLE  ALT=\"15-2%281.5%29+=+15-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >inches.<BR>\n" );
document.write( "Width, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+7-2x ALIGN=MIDDLE  ALT=\"W+=+7-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-2%281.5%29+=+7-3 ALIGN=MIDDLE  ALT=\"7-2%281.5%29+=+7-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >inches.<BR>\n" );
document.write( "Height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+x ALIGN=MIDDLE  ALT=\"h+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5 ALIGN=MIDDLE  ALT=\"1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >inches.<BR>\n" );
document.write( "The maximum volume of the box would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bmax%5D+=+L%2AW%2Ah ALIGN=MIDDLE  ALT=\"V%5Bmax%5D+=+L%2AW%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bmax%5D+=+12%2A4%2A1.5 ALIGN=MIDDLE  ALT=\"V%5Bmax%5D+=+12%2A4%2A1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bmax%5D+=+72 ALIGN=MIDDLE  ALT=\"V%5Bmax%5D+=+72\"   DISABLED_style= \"max-width:90%; height:auto;\" >cubic inches.<BR>\n" );
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