document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=201030>Question 201030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> not sure if in right area but can someone help me with this problem\r<BR>\n" );
document.write( "\n" );
document.write( "An open box is to be constructed from a piece of cardboard that is 30in. by 30in. by cutting a square out of each corner and folding up the sides.  What are the dimensions of the box that will yield the maximum volume?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so confused<BR>\n" );
document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #151273 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=151273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is no correct area on this site for this problem because it is a Calculus local maximum problem.  The closest thing I could find was <i>Algebra: Polynomials, rational expressions, and equations</i>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> represent the measure of the side of the squares cut out of the corners.  That means that the width and length of the resulting box must measure <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 30 - 2x\"> and the height of the resulting box is just <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  So you can write a function representing the volume of the box in terms of the dimension of the cutout square thus:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x) = x(30 - 2x)(30 - 2x) = 4x^3 - 120x^2 + 900x\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given a continuous and differentiable function, such as the above polynomial function, local extrema can be found at the value of the independent variable for which the value of the first derivative is equal to zero.  So, take the first derivative:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx} = 12x^2 - 240x + 900\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Set the first derivative equal to zero and solve:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 12x^2 - 240x + 900 = 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (3x - 45)(4x - 20) = 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hence, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinates of the local extrema are 5 and 15.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given a continuous and twice-differentiable function if the 2nd derivative is less than zero evaluated at the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of an extrema, that extrema is a maximum.  If it is positive, the extrema is a minimum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take the 2nd derivative:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2} = 24x - 240\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then evaluate at 5:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V''(5) = 24(5) - 240 = -120\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So 5 is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of a maximum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V''(15) = 24(15) - 240 = 120\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So 15 is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of a minimum.  Which, by the way, makes very good sense because if you cut 15 inch squares out of the corners of a 30 by 30 piece of material, you would end up with a 15 X 0 X 0 box having zero volume.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, you cut 5 inch squares from each corner, leaving you with a 5 X 20 X 20 box.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's a picture of your volume function:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-1%2C+17%2C+-100%2C+2100%2C%0D%0Agrid%281%29%2C%0D%0Agreen%28circle%285%2C2000%2C.2%29%29%2C%0D%0Agraph%28%0D%0A500%2C+500%2C+-1%2C+17%2C+-100%2C+2100%2C%0D%0A4x%5E3-120x%5E2%2B900x%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );