document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166560>Question 166560</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hey, I just need help with two word problems.<BR>\n" );
document.write( "1st one:   A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg?? \r<BR>\n" );
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document.write( "2nd one:   A rectangular piece of cardboard measuring 28 inches by 29 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. For what value of x will the volume be a maximum? If necessary, round to 2 decimal places. \r<BR>\n" );
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document.write( "Thank u  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122757 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=122757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1st one: A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg??<BR>\n" );
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document.write( "Let x = length of longer leg<BR>\n" );
document.write( "then<BR>\n" );
document.write( "x+60 = hypotenuse<BR>\n" );
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document.write( "Applying pythagorean's theorem:<BR>\n" );
document.write( "180^2 + x^2 = (x+60)^2<BR>\n" );
document.write( "180^2 + x^2 = (x+60)^2<BR>\n" );
document.write( "32400 + x^2 = x^2 + 120x + 3600<BR>\n" );
document.write( "32400 = 120x + 3600<BR>\n" );
document.write( "28800 = 120x<BR>\n" );
document.write( "240 meters = x (longer leg)<BR>\n" );
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document.write( "2nd one: A rectangular piece of cardboard measuring 28 inches by 29 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. For what value of x will the volume be a maximum? If necessary, round to 2 decimal places.<BR>\n" );
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document.write( "Volume = x(28-2x)(29-2x)<BR>\n" );
document.write( "Volume = x(812-56x-58x+4x^2)<BR>\n" );
document.write( "Volume = x(812-114x+4x^2)<BR>\n" );
document.write( "Volume = 812x-114x^2+4x^3<BR>\n" );
document.write( "Volume = 4x^3-114x^2+812x<BR>\n" );
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document.write( "Taking the derivative:<BR>\n" );
document.write( "4x^3-114x^2+812x <BR>\n" );
document.write( "12x^2-228x+812 = 0<BR>\n" );
document.write( "6x^2-114x+406 = 0<BR>\n" );
document.write( "Solving the above with the quadratic equation we get:<BR>\n" );
document.write( "x = {14.25, 4.75}<BR>\n" );
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document.write( "Solution: 4.75 inches<BR>\n" );
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document.write( "Details of quadratic equation:<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=6x%5E2%2B-114x%2B406+=+0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-114x%2B406+=+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=%28-114%29%5E2-4%2A6%2A406=3252 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-114%29%5E2-4%2A6%2A406=3252\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=3252 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--114%2B-sqrt%28+3252+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--114%2B-sqrt%28+3252+%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-%28-114%29%2Bsqrt%28+3252+%29%29%2F2%5C6+=+14.2521924764611 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-114%29%2Bsqrt%28+3252+%29%29%2F2%5C6+=+14.2521924764611\"   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-%28-114%29-sqrt%28+3252+%29%29%2F2%5C6+=+4.74780752353892 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-114%29-sqrt%28+3252+%29%29%2F2%5C6+=+4.74780752353892\"   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=6x%5E2%2B-114x%2B406 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-114x%2B406\"   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=6x%5E2%2B-114x%2B406+=+6%28x-14.2521924764611%29%2A%28x-4.74780752353892%29 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-114x%2B406+=+6%28x-14.2521924764611%29%2A%28x-4.74780752353892%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 14.2521924764611, 4.74780752353892.\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+6%2Ax%5E2%2B-114%2Ax%2B406+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-114%2Ax%2B406+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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