document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=800985>Question 800985</A>: <I><SPAN STYLE=\"word-break: break-all;\"> POLYNOMIAL BOX\r<BR>\n" );
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document.write( "Let's say that I have a 50cm x 60cm cardboard. My problem/task is to design a box (open top) that will get the greatest capacity or maximum volume of the cardboard can get. Thus, the size of the square to be cut will be my main problem such that the box will have a maximum capacity or volume. Each side of the square is represented by x. So my question is, what is the value of x so that the volume of the box will reach its maximum?\r<BR>\n" );
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document.write( "OPEN TOP<BR>\n" );
document.write( "Height = (x)<BR>\n" );
document.write( "Length = 60-2x<BR>\n" );
document.write( "Width = 50-2x\r<BR>\n" );
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document.write( "Since, V=Length x Width x Height<BR>\n" );
document.write( "P(x)= (60-2x)(50-2x)(x)\r<BR>\n" );
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document.write( "Thanks in Advance! :).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #483396 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=483396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's say that I have a 50cm x 60cm cardboard. My problem/task is to design a box (open top) that will get the greatest capacity or maximum volume of the cardboard can get. Thus, the size of the square to be cut will be my main problem such that the box will have a maximum capacity or volume. Each side of the square is represented by x. So my question is, what is the value of x so that the volume of the box will reach its maximum?\r<BR>\n" );
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document.write( "OPEN TOP<BR>\n" );
document.write( "Height = (x)<BR>\n" );
document.write( "Length = 60-2x<BR>\n" );
document.write( "Width = 50-2x\r<BR>\n" );
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document.write( "Since, V=Length x Width x Height<BR>\n" );
document.write( "P(x)= (60-2x)(50-2x)(x)<BR>\n" );
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document.write( "P(x) = 4x^3 - 220x^2 + 3000x<BR>\n" );
document.write( "Find the max of the function.<BR>\n" );
document.write( "P'(x) = 12x^2 - 440x + 3000 = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+110x+%2B+750+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2+-+110x+%2B+750+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B-110x%2B750+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-110x%2B750+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-110%29%5E2-4%2A3%2A750=3100 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-110%29%5E2-4%2A3%2A750=3100\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=3100 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--110%2B-sqrt%28+3100+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--110%2B-sqrt%28+3100+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-110%29%2Bsqrt%28+3100+%29%29%2F2%5C3+=+27.6129406047167 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-110%29%2Bsqrt%28+3100+%29%29%2F2%5C3+=+27.6129406047167\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-110%29-sqrt%28+3100+%29%29%2F2%5C3+=+9.05372606194996 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-110%29-sqrt%28+3100+%29%29%2F2%5C3+=+9.05372606194996\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B-110x%2B750 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-110x%2B750\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B-110x%2B750+=+%28x-27.6129406047167%29%2A%28x-9.05372606194996%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-110x%2B750+=+%28x-27.6129406047167%29%2A%28x-9.05372606194996%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 27.6129406047167, 9.05372606194996.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-110%2Ax%2B750+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-110%2Ax%2B750+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "x = ~ 9.04 gives a volume of ~12096 cc<BR>\n" );
document.write( "The 27... value is a local minimum.<BR>\n" );
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document.write( "The function is not factorable, so I don't see a way to do it without using the derivative.<BR>\n" );
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