document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=309225>Question 309225</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The height (h) of a baseball (t) seconds after being hit is given by h(t)= -16t^2 + 80t + 9. What is the height of the baseball when it is into the air? What is the maximum height of the baseball and when does it reach this height? When does the baseball hit the ground? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #221188 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=221188\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">     The height (h) of a baseball (t) seconds after being hit is given by h(t)= -16t^2 + 80t + 9. <BR>\n" );
document.write( "What is the height of the baseball when it is into the air? <BR>\n" );
document.write( "The height is given by the equation:<BR>\n" );
document.write( "h(t)= -16t^2 + 80t + 9<BR>\n" );
document.write( "where<BR>\n" );
document.write( "h(t) is the height <BR>\n" );
document.write( "t is time in seconds<BR>\n" );
document.write( ".<BR>\n" );
document.write( "What is the maximum height of the baseball and when does it reach this height? <BR>\n" );
document.write( "Maximum height is at the vertex:<BR>\n" );
document.write( "t = -b/(2a) =  -80/(2*(-16)) = 80/32 =  2.5<BR>\n" );
document.write( "h(2.5) =  -16(2.5^2) + 80(2.5) + 9 = 109 feet<BR>\n" );
document.write( "When does the baseball hit the ground?<BR>\n" );
document.write( "set h(t) to zero solve for t:<BR>\n" );
document.write( "0 = -16t^2 + 80t + 9<BR>\n" );
document.write( "Solve using the quadratic formula to get:<BR>\n" );
document.write( "t = {-0.11, 5.11}<BR>\n" );
document.write( "Throw out the negative solution leaving:<BR>\n" );
document.write( "t = 5.11 secs<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Details of quadratic :<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=-16x%5E2%2B80x%2B9+=+0 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B80x%2B9+=+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=%2880%29%5E2-4%2A-16%2A9=6976 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2880%29%5E2-4%2A-16%2A9=6976\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=6976 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+=+%28-80%2B-sqrt%28+6976+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-80%2B-sqrt%28+6976+%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-%2880%29%2Bsqrt%28+6976+%29%29%2F2%5C-16+=+-0.110076627227638 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2880%29%2Bsqrt%28+6976+%29%29%2F2%5C-16+=+-0.110076627227638\"   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-%2880%29-sqrt%28+6976+%29%29%2F2%5C-16+=+5.11007662722764 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2880%29-sqrt%28+6976+%29%29%2F2%5C-16+=+5.11007662722764\"   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=-16x%5E2%2B80x%2B9 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B80x%2B9\"   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=-16x%5E2%2B80x%2B9+=+-16%28x--0.110076627227638%29%2A%28x-5.11007662722764%29 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B80x%2B9+=+-16%28x--0.110076627227638%29%2A%28x-5.11007662722764%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.110076627227638, 5.11007662722764.\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+-16%2Ax%5E2%2B80%2Ax%2B9+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-16%2Ax%5E2%2B80%2Ax%2B9+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );