document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=147162>Question 147162</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For 1989 and 1990 Dave Johnson had the highest decathlon score in the world.   When Johnson reached a speed of 32 ft/sec on the pole vault runway, his height above the ground t seconds after leaving the ground was given <BR>\n" );
document.write( "by h= -16t^2+32t  .  (The elasticity of the pole converts the horizontal speed into vertical speed.)  Determine how long Johnson was in the air.  For how long was he more than 14 ft in the air?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #107509 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=107509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> h= -16t^2+32t<BR>\n" );
document.write( "The problem asks how long he was ABOVE 14 feet -- which is h (height).  Simply plug it into the equation and solve for t (time).<BR>\n" );
document.write( ".<BR>\n" );
document.write( "h= -16t^2+32t<BR>\n" );
document.write( "14 = -16t^2+32t<BR>\n" );
document.write( "Move everything to the left side:<BR>\n" );
document.write( "16t^2 - 32t + 14 = 0<BR>\n" );
document.write( "Divide by throughout by 2:<BR>\n" );
document.write( "8t^2 - 16t + 7 = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since you can't factor, you must use the quadratic equation:<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=8x%5E2%2B-16x%2B7+=+0 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-16x%2B7+=+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=%28-16%29%5E2-4%2A8%2A7=32 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-16%29%5E2-4%2A8%2A7=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=32 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--16%2B-sqrt%28+32+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--16%2B-sqrt%28+32+%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-%28-16%29%2Bsqrt%28+32+%29%29%2F2%5C8+=+1.35355339059327 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-16%29%2Bsqrt%28+32+%29%29%2F2%5C8+=+1.35355339059327\"   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-%28-16%29-sqrt%28+32+%29%29%2F2%5C8+=+0.646446609406726 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-16%29-sqrt%28+32+%29%29%2F2%5C8+=+0.646446609406726\"   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=8x%5E2%2B-16x%2B7 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-16x%2B7\"   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=8x%5E2%2B-16x%2B7+=+8%28x-1.35355339059327%29%2A%28x-0.646446609406726%29 ALIGN=MIDDLE  ALT=\"8x%5E2%2B-16x%2B7+=+8%28x-1.35355339059327%29%2A%28x-0.646446609406726%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.35355339059327, 0.646446609406726.\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+8%2Ax%5E2%2B-16%2Ax%2B7+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B-16%2Ax%2B7+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "x has two solutions:<BR>\n" );
document.write( "1.35355339059327, 0.646446609406726<BR>\n" );
document.write( "The difference is the time he was above 14 feet:<BR>\n" );
document.write( "1.35355339059327 - 0.646446609406726 = .0607 seconds<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );