document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=664592>Question 664592</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the maximum volume of a open box created from a square of side length 24? Keep in mind that V=LWH. So for this problem the dimensions are as follows: <BR>\n" );
document.write( "L = 24-2x, W = 24-2x and H = x. The maximum volume will be where the volume equation is the greatest.<BR>\n" );
document.write( "You are to determine: <BR>\n" );
document.write( "1. The volume equation. <BR>\n" );
document.write( "2. The maximum volume. <BR>\n" );
document.write( "3. The value of x that maximizes the volume. <BR>\n" );
document.write( " Explain each step. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #413373 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=413373\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Multiply length times width times height:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(24\ -\ 2x\right)^2\left(x\right)\ =\ \left(4x^2\ -\ 96x\ +\ 576\right)\left(x\right)\ =\ 4x^3\ -\ 96x^2\ +\ 576x\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ 4x^3\ -\ 96x^2\ +\ 576x\">\r<BR>\n" );
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document.write( "Take the first derivative (Power and Sum Rules)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ 12x^2\ -\ 192x\ +\ 576\">\r<BR>\n" );
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document.write( "Set the first derivative equal to zero and solve.  This is a quadratic so you will get two roots, each of which represents the abscissa of a local extremum of the original function.\r<BR>\n" );
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document.write( "Take the second derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ 24x\ -\ 192\">\r<BR>\n" );
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document.write( "Evaluate\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f^{\left[2\right]}(a) \">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> is either of the abscissas of the possible extreme points.   If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{\left[2\right]}(a) \ <\ 0\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> is the abscissa of a local maximum, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{\left[2\right]}(a)\ >\ 0\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> is the abscissa of a local minimum.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{\left[2\right]}(a)\ =\ 0\">, the second derivative test is inconclusive, but it is a possible inflection point.   In the latter case, for this problem, calculate the volume for the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> in question and see if it makes sense as a possible <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> value that could give a maximum volume.\r<BR>\n" );
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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( "My calculator said it, I believe it, that settles it<BR>\n" );
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