document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86483>Question 86483</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and 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" );
document.write( "Answer: \r<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.\r<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.  <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62592 by </B><A HREF=/tutors/aboutme.mpl?userid=Nate><B>Nate(3500)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Nate><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=62592\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and 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" );
document.write( "Answer:<BR>\n" );
document.write( "Drawing a picture would be extremely helpful.<BR>\n" );
document.write( "1. Draw a 4 by 6 foot rectangle<BR>\n" );
document.write( "2. Make boxes at each of the four corners with sides x feet<BR>\n" );
document.write( "Length: 6 - 2x<BR>\n" );
document.write( "Width: 4 - 2x<BR>\n" );
document.write( "Height: x<BR>\n" );
document.write( "V = product of all sides<BR>\n" );
document.write( "V = (6 - 2x)(4 - 2x)(x)<BR>\n" );
document.write( "V = 4(3 - x)(2 - x)(x)<BR>\n" );
document.write( "V = 4(6 - 5x + x^2)(x)<BR>\n" );
document.write( "V = 4(6x - 5x^2 + x^3)<BR>\n" );
document.write( "V = 24x - 20x^2 + 4x^3\r<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. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C4%2C-10%2C10%2C24x+-+20x%5E2+%2B+4x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C4%2C-10%2C10%2C24x+-+20x%5E2+%2B+4x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<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.<BR>\n" );
document.write( "Approx. when x = 1<BR>\n" );
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document.write( "V(x) = 24x - 20x^2 + 4x^3<BR>\n" );
document.write( "V'(x) = 24 - 40x + 12x^2<BR>\n" );
document.write( "0 = 6 - 10x + 3x^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28+100+-+72+%29%29%2F%286%29+ ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28+100+-+72+%29%29%2F%286%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28+28+%29%29%2F%286%29+ ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28+28+%29%29%2F%286%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+2%2Asqrt%28+7+%29%29%2F%286%29+ ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+2%2Asqrt%28+7+%29%29%2F%286%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+%2B-+sqrt%28+7+%29%29%2F%283%29+ ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%28+7+%29%29%2F%283%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "~<BR>\n" );
document.write( "Exact: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+-+sqrt%287%29%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%285+-+sqrt%287%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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