document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112462>Question 112462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How Would I create a Real Life Example Involving Polynomials? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82046 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82046\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's say you have a square piece of sheet aluminum that is 12 inches on each side.  You want to cut a square of dimension x by x from each corner of the sheet and then fold the sides up to make an open topped box.  What dimension should you select for x so that box you make has the maximum possible volume?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the original sheet of metal is 12\" by 12\", and you cut an x by x square from each corner, the dimensions of the bottom of the box would then be 12 - 2x inches.  Then the volume with respect to the x dimension of the box would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=%2812-2x%29%2812-2x%29x ALIGN=MIDDLE  ALT=\"V%28x%29=%2812-2x%29%2812-2x%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First, let's test a couple of values to see if it makes any difference.  Let's try x = 1.  x = 1 means that our box would be 10 by 10 on the bottom and 1 inch tall for a total volume of 100 cubic inches.  Let's try x = 3.  x = 3 means that the box would be 6 by 6 on the bottom and 3 inches tall, 6 X 6 X 3 = 108 cubic inches.  So clearly the selected value for x makes a difference in the volume.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=4x%5E3-48x%5E2%2B144x ALIGN=MIDDLE  ALT=\"V%28x%29=4x%5E3-48x%5E2%2B144x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the polynomial function, and the real life problem would be to find the value of x that makes the volume a maximum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-1%2C12%2C-20%2C150%2C4x%5E3-48x%5E2%2B144x%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-1%2C12%2C-20%2C150%2C4x%5E3-48x%5E2%2B144x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph illustrates the situation.  Values for x larger than 6 don't have any application because if x were 6 or larger, there wouldn't be any metal left to make a box.  So the interval of interest is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C6 ALIGN=MIDDLE  ALT=\"0%3Cx%3C6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Ok.  Extra credit.  It looks like the x value at the maximum is about 2.   What is the volume of the box if x = 2? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );