document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200116>Question 1200116</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An open box is to be constructed from a 12 x 12 inch piece of board by cutting away squares of equal size from the four corners and folding up the sides. Determine the size of the cut-out that maximizes the volume of the box?<BR>\n" );
document.write( "A. 350 <BR>\n" );
document.write( "B. 274<BR>\n" );
document.write( "C. 231<BR>\n" );
document.write( "D. 128 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834147 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Starting with the 12 x 12 box, let's call the length of the cutout, X.<BR>\n" );
document.write( "So then the width of the side becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=12-2X ALIGN=MIDDLE  ALT=\"L=12-2X\"   DISABLED_style= \"max-width:90%; height:auto;\" > since there are two cutouts. Since it's a square cutout off of a square, the height and width are the same.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=X.jpg\"><BR>\n" );
document.write( "When you fold up the open top box, you get the dimensions you need to find the volume of the box.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=BOX\"><BR>\n" );
document.write( "So then the volume of the box becomes,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%2812-2X%29%2812-2X%29X ALIGN=MIDDLE  ALT=\"V=%2812-2X%29%2812-2X%29X\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%284X%5E2-48X%2B144%29X ALIGN=MIDDLE  ALT=\"V=%284X%5E2-48X%2B144%29X\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=4X%5E3-48X%5E2%2B144X ALIGN=MIDDLE  ALT=\"V=4X%5E3-48X%5E2%2B144X\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since you have the volume as a function of one variable, you can take the derivative and set it equal to zero to find the extrema. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2FdX=12X%5E2-96X%2B144=0 ALIGN=MIDDLE  ALT=\"dV%2FdX=12X%5E2-96X%2B144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2-8X%2B12=0 ALIGN=MIDDLE  ALT=\"X%5E2-8X%2B12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28X-6%29%28X-2%29=0 ALIGN=MIDDLE  ALT=\"%28X-6%29%28X-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So there are two solutions, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X=6 ALIGN=MIDDLE  ALT=\"X=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X=2 ALIGN=MIDDLE  ALT=\"X=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "You can use the second derivative test to find which value gives you the maximum  or the minimum. <BR>\n" );
document.write( "As an alternative, you can also just plug the value into the volume equation.<BR>\n" );
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document.write( "In this case, neither value matches any of your choices. So I presume the choices are the maximum volume and not the size of the cutout. You can verify. </SPAN>\n" );
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