document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=71147>Question 71147</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangle is 32\", and the area is 60 square inches. Find the length and width of the rectangle. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #50869 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=50869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let L=length and h=height. In this case the equations for perimeter and area are:<BR>\n" );
document.write( "2L+2h=32 and L*h=60<BR>\n" );
document.write( "L=60/h<BR>\n" );
document.write( "Now plug in L=60/h for L in the previous equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2860%2Fh%29%2B2h=32 ALIGN=MIDDLE  ALT=\"2%2860%2Fh%29%2B2h=32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%2Fh%2B2h%5E2%2Fh=32 ALIGN=MIDDLE  ALT=\"120%2Fh%2B2h%5E2%2Fh=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >Multiply 2nd term on left side to reach common denominator<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28120%2B2h%5E2%29%2Fh=32 ALIGN=MIDDLE  ALT=\"%28120%2B2h%5E2%29%2Fh=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >Multiply both sides by h<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28120%2B2h%5E2%29=32h ALIGN=MIDDLE  ALT=\"%28120%2B2h%5E2%29=32h\"   DISABLED_style= \"max-width:90%; height:auto;\" >Get everything to one side by subtraction<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2h%5E2-32h%2B120=0 ALIGN=MIDDLE  ALT=\"2h%5E2-32h%2B120=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >Now we have a quadratic to solve. Use the quadratic equation to solve.\r<BR>\n" );
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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=ah%5E2%2Bbh%2Bc=0 ALIGN=MIDDLE  ALT=\"ah%5E2%2Bbh%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2h%5E2%2B-32h%2B120+=+0 ALIGN=MIDDLE  ALT=\"2h%5E2%2B-32h%2B120+=+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=h%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"h%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=%28-32%29%5E2-4%2A2%2A120=64 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-32%29%5E2-4%2A2%2A120=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=64 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--32%2B-sqrt%28+64+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--32%2B-sqrt%28+64+%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=h%5B1%5D+=+%28-%28-32%29%2Bsqrt%28+64+%29%29%2F2%5C2+=+10 ALIGN=MIDDLE  ALT=\"h%5B1%5D+=+%28-%28-32%29%2Bsqrt%28+64+%29%29%2F2%5C2+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B2%5D+=+%28-%28-32%29-sqrt%28+64+%29%29%2F2%5C2+=+6 ALIGN=MIDDLE  ALT=\"h%5B2%5D+=+%28-%28-32%29-sqrt%28+64+%29%29%2F2%5C2+=+6\"   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=2h%5E2%2B-32h%2B120 ALIGN=MIDDLE  ALT=\"2h%5E2%2B-32h%2B120\"   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=2h%5E2%2B-32h%2B120+=+2%28h-10%29%2A%28h-6%29 ALIGN=MIDDLE  ALT=\"2h%5E2%2B-32h%2B120+=+2%28h-10%29%2A%28h-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10, 6.\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+2%2Ax%5E2%2B-32%2Ax%2B120+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-32%2Ax%2B120+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " <BR>\n" );
document.write( "So the height is either 6 or 10. Lets solve for L.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10L=60 ALIGN=MIDDLE  ALT=\"10L=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=60%2F10 ALIGN=MIDDLE  ALT=\"L=60%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=6 ALIGN=MIDDLE  ALT=\"L=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >If h=10 then L=6<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6L=60 ALIGN=MIDDLE  ALT=\"6L=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=60%2F6 ALIGN=MIDDLE  ALT=\"L=60%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=6 ALIGN=MIDDLE  ALT=\"L=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >If h=6 then L=10, see a pattern? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );