document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=421518>Question 421518</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions of 6 in by 10 in. The metal is then folded to make a open top box. What is the maximum volume of the box and what are its dimensions.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #294385 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=294385\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the length of the box will be 10 - 2x<BR>\n" );
document.write( "the width of the bot will be 6 - 2x<BR>\n" );
document.write( "the height of the box will be x\r<BR>\n" );
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document.write( "the volume of the box will be (10 - 2x) * (6 - 2x) * x\r<BR>\n" );
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document.write( "the value of x has to be less than 3 since if it was equal to 3, then the width of the box would be equal to 0.\r<BR>\n" );
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document.write( "the value of x can be 0, so we have a domain where 0 <= x < 3.\r<BR>\n" );
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document.write( "if we multiply all factors together, we get:\r<BR>\n" );
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document.write( "y = 4x^3 - 32x^2 + 60x\r<BR>\n" );
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document.write( "if we graph this equation, we get:\r<BR>\n" );
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document.write( "within our domain space, it looks like we will get a maximum value of y at somewhere around y = 30 when x is somewhere around x = 1.\r<BR>\n" );
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document.write( "to solve for the roots of this equation, we set it equal to 0.\r<BR>\n" );
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document.write( "we get 4x^3 - 32x^2 + 60x = 0\r<BR>\n" );
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document.write( "we factor out an x and a 4 to get:\r<BR>\n" );
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document.write( "4 * x * (x^2 - 8x + 15) = 0\r<BR>\n" );
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document.write( "this equation will have roots at x = 0, x = 3, x = 5\r<BR>\n" );
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document.write( "the root at x = 5 is outside our domain of 0 <= x < 3, so we are left with roots at x = 0 and x = 3\r<BR>\n" );
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document.write( "to find the max / min point of this cubic equation, we have to resort to calculus (i don't know any other way).\r<BR>\n" );
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document.write( "the equation is y = 4x^3 - 32x^2 + 60x\r<BR>\n" );
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document.write( "the derivative of the equation with respect to x is:\r<BR>\n" );
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document.write( "dy/dx = 12x^2 - 64x + 60\r<BR>\n" );
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document.write( "this derivative is the slope of the equation at a particular value of x.\r<BR>\n" );
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document.write( "we want the slope of the equation to be equal to 0 which would indicate a max / min point of the original equation.\r<BR>\n" );
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document.write( "we set dy/dx equal to 0 to get:\r<BR>\n" );
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document.write( "12x^2 - 64x + 60 = 0\r<BR>\n" );
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document.write( "since this is a standard quadratic equation in the form of ax^2 + bx + 2 = 0, we can solve for the roots of this equation by setting:\r<BR>\n" );
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document.write( "a = 12<BR>\n" );
document.write( "b = -64<BR>\n" );
document.write( "c = 60\r<BR>\n" );
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document.write( "from that, we derive the piece parts of the quadratic formula of:\r<BR>\n" );
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document.write( "2a = 24<BR>\n" );
document.write( "-b = 64<BR>\n" );
document.write( "b^2 = 4096<BR>\n" );
document.write( "4ac = 2880<BR>\n" );
document.write( "b^2 - 4ac = 2880\r<BR>\n" );
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document.write( "we then plug these values into the quadratic formula of x = (-b +/- sqrt(b^2-4ac))/2a to get:\r<BR>\n" );
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document.write( "x = (64 +/- sqrt(2880))/24\r<BR>\n" );
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document.write( "this gets us x = 4.119632981 or x = 1.213700352\r<BR>\n" );
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document.write( "x = 4.... is outside our domain so we are left with x = 1.213700352.\r<BR>\n" );
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document.write( "when x = 1.213700352, the equation of y = 4x^3 - 32x^2 + 60x becomes y = 32.83528294.\r<BR>\n" );
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document.write( "if we add a value of y = 32.83528294 to our equation that we previously graphed, we should see that it is at the maximum value of the equation, as shown below:\r<BR>\n" );
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document.write( "the solution to your problem is that the maximum volume of the rectangular solid is 32.83528294 cubic inches.\r<BR>\n" );
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document.write( "this occurs when the value of x is 1.213700352 inches.\r<BR>\n" );
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document.write( "i'm far from a calculus guru, but i knew enough to solve this problem using it.\r<BR>\n" );
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document.write( "hopefully this is in a class where you use calculus.\r<BR>\n" );
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document.write( "i didn't know how to solve for the maximum volume any other way.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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