document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=985067>Question 985067</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box is to be constructed from a sheet of cardboard that is 20 cm by 50 cm by cutting out squares of length x by x from each corner and bending up the sides. \r<BR>\n" );
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document.write( "What is the maximum volume this box could have? (Round your answer to two decimal places. Do not include units, for example, 10.22 cm would be 10.22.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #605902 by </B><A HREF=/tutors/aboutme.mpl?userid=macston><B>macston(5194)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=macston><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=macston><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=605902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "L=length=50-2x; W=width=20-2x; H=height=x; V=volume<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=LWH ALIGN=MIDDLE  ALT=\"V=LWH\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%2850-2x%29%2820-2x%29%28x%29 ALIGN=MIDDLE  ALT=\"V=%2850-2x%29%2820-2x%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=4x%5E3-140x%5E2%2B1000x ALIGN=MIDDLE  ALT=\"V=4x%5E3-140x%5E2%2B1000x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The function will have a maximum at the point where the first derivative equals zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx=12x%5E2-280x%2B1000 ALIGN=MIDDLE  ALT=\"dV%2Fdx=12x%5E2-280x%2B1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-280x%2B1000=0 ALIGN=MIDDLE  ALT=\"12x%5E2-280x%2B1000=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-70x%2B250=0 ALIGN=MIDDLE  ALT=\"3x%5E2-70x%2B250=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<TABLE  BORDER=1>\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=quadratic.solver>SOLVE quadratic equation with variable</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-70x%2B250+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-70x%2B250+=+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=%28-70%29%5E2-4%2A3%2A250=1900 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-70%29%5E2-4%2A3%2A250=1900\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1900 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--70%2B-sqrt%28+1900+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--70%2B-sqrt%28+1900+%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-%28-70%29%2Bsqrt%28+1900+%29%29%2F2%5C3+=+18.9314982392345 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-70%29%2Bsqrt%28+1900+%29%29%2F2%5C3+=+18.9314982392345\"   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-%28-70%29-sqrt%28+1900+%29%29%2F2%5C3+=+4.40183509409888 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-70%29-sqrt%28+1900+%29%29%2F2%5C3+=+4.40183509409888\"   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-70x%2B250 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-70x%2B250\"   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-70x%2B250+=+3%28x-18.9314982392345%29%2A%28x-4.40183509409888%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-70x%2B250+=+3%28x-18.9314982392345%29%2A%28x-4.40183509409888%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 18.9314982392345, 4.40183509409888.\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-70%2Ax%2B250+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-70%2Ax%2B250+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( ".<BR>\n" );
document.write( "Determine the domain of the function. x must be greater than 0 or H would be zero and we would have a flat sheet, so x>0. 2x must be less than 20 or we would have no width so 2x<20 or x<10<BR>\n" );
document.write( "So 0 < x < 10.<BR>\n" );
document.write( "That leaves us with x=4.40 as the solution.<BR>\n" );
document.write( "The size of the box:<BR>\n" );
document.write( "L=50-2x=50-8.80=41.20<BR>\n" );
document.write( "W=20-2x=20-8.80=11.20<BR>\n" );
document.write( "H=x=4.40<BR>\n" );
document.write( "Maximum Volume of the box=(41.20)(11.20)(4.40)=2030.34 cm^3 \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );