document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=229619>Question 229619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> n open box will be made out of a piece of metal that is 24 inches by 30 inches by cutting squares out of the corners and folding up the sides. Write an equation describing the volume of the box in terms of x and find how big we should cut the squares to maximize the volume of the box. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170226 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=170226\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Starting with the 24\" X 30\" piece of metal, remove the four corners by cutting out x by x squares.<BR>\n" );
document.write( "The volume of the box made by folding up the sides can be expressed by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2824-2x%29%2A%2830-2x%29%2Ax ALIGN=MIDDLE  ALT=\"V+=+%2824-2x%29%2A%2830-2x%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > where x = the height of the newly-formed box, (24-2x) = the width of the base, and (30-2x) = the length of the base.<BR>\n" );
document.write( "Now that we have the equation for the volume of the box<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+4x%5E3-108x%5E2%2B720x ALIGN=MIDDLE  ALT=\"V+=+4x%5E3-108x%5E2%2B720x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can find its maximum value by differentating V with respect to x and setting this equal to zero.<BR>\n" );
document.write( "What we are really doing, of course, is finding the points on the cubic curve at which the slope is zero, thus we are findinding the relative maximum and relative minimum points on this cubic curve.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+4x%5E3-108x%5E2%2B720x ALIGN=MIDDLE  ALT=\"V+=+4x%5E3-108x%5E2%2B720x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Differentiate.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+12x%5E2-216x%2B420 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+12x%5E2-216x%2B420\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-216x%2B720+=+0 ALIGN=MIDDLE  ALT=\"12x%5E2-216x%2B720+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify by factoring a 12.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28x%5E2-18x%2B60%29+=+0 ALIGN=MIDDLE  ALT=\"12%28x%5E2-18x%2B60%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so, by the zero product rule, we have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-18x%2B60+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-18x%2B60+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since this won't factor, we'll use the quadratic formula to solve for x. <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 = 1, b = -18, and c = 60.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%2818%29%2B-sqrt%28%28-18%29%5E2-4%281%29%2860%29%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%2818%29%2B-sqrt%28%28-18%29%5E2-4%281%29%2860%29%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+13.58%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+13.58%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28x+=+4.42%29 ALIGN=MIDDLE  ALT=\"highlight_green%28x+=+4.42%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Notice that the larger solution is of no practical value since, if the cut-out squares are 13.58\" by 13.58\", this would take away more material than you have in the original piece of metal!<BR>\n" );
document.write( "So, to obtain the maximum volume for the box, the squares must measure 4.42\" by 4.42\"<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C20%2C-130%2C1500%2C4x%5E3-108x%5E2%2B720x%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C20%2C-130%2C1500%2C4x%5E3-108x%5E2%2B720x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );