document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=38205>Question 38205</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)	An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.<BR>\n" );
document.write( "a)	Find the function V that represents the volume of the box in terms of x.<BR>\n" );
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document.write( "b)	Graph this function and show the graph over the valid range of the variable x..<BR>\n" );
document.write( "Show Graph here.<BR>\n" );
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document.write( "c)	Using the graph, what is the value of x that will produce the maximum volume?<BR>\n" );
document.write( "Answer.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #23699 by </B><A HREF=/tutors/aboutme.mpl?userid=fractalier><B>fractalier(6550)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=fractalier><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=fractalier><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=23699\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Again,<BR>\n" );
document.write( "If we cut out x from all four corners, the new length will be 6 - 2x and the new width will be 4 - 2x, with the height of the box being just x.<BR>\n" );
document.write( "Now Volume V = lwh, so we have<BR>\n" );
document.write( "V = x(6 - 2x)(4 - 2x)<BR>\n" );
document.write( "I cannot graph it for you, but you can see that x must be more than zero and less than 2.<BR>\n" );
document.write( "Using early calculus I can show you how to find the maximum area, but without a way to show you it's graph, that's the best I can do...<BR>\n" );
document.write( "We need to maximize the function<BR>\n" );
document.write( "V(x) = x(6 - 2x)(4 - 2x)<BR>\n" );
document.write( "We do that by taking its derivative and setting it equal to zero, then solving for x...here goes...<BR>\n" );
document.write( "V(x) = x(6 - 2x)(4 - 2x)<BR>\n" );
document.write( "V(x) = 24x - 20x^2 + 4x^3<BR>\n" );
document.write( "V'(x) = 24 - 40x + 12x^2 thus<BR>\n" );
document.write( "12x^2 - 40x + 24 = 0<BR>\n" );
document.write( "3x^2 - 10x + 6 = 0<BR>\n" );
document.write( "This isn't factorable so we use the quadratic formula and get<BR>\n" );
document.write( "x = (10 ± 2sqrt(7)) / 6 = (5 ± sqrt(7)) / 3<BR>\n" );
document.write( "The positive root is too big, so x must be<BR>\n" );
document.write( "x = (5 - sqrt(7)) / 3 or about .785<BR>\n" );
document.write( "Now plug that in to V(x) to get the maximum volume...<BR>\n" );
document.write( "V(.785) = .785(6 - 2(.785))(4 - 2(.785)) = 8.45 cubic feet </SPAN>\n" );
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