document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=213153>Question 213153</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A baseball is thrown upward with an initial speed of 69ft/sec. The number of feet s above the ground after t seconds is given by the equation s=-16t^6+69t. When will the baseball be 54 feet above 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 #161050 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=161050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem gives you:<BR>\n" );
document.write( "s=-16t^6+69t<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Where s = feet above ground<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now, they would like to know at what time (t) is it when it is 54 feet above the ground.  Simply substitute s for 54 and solve for t:<BR>\n" );
document.write( "s=-16t^6+69t<BR>\n" );
document.write( "54=-16t^6+69t<BR>\n" );
document.write( "0=-16t^6+69t-54<BR>\n" );
document.write( "0=16t^6-69t+54<BR>\n" );
document.write( "Solve by using the quadratic formula.  Doing so yields:<BR>\n" );
document.write( "t = {3.285, 1.027} (in seconds)<BR>\n" );
document.write( "The reason for two answers -- once going up and another time coming down.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Details of quadratic below:<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=at%5E2%2Bbt%2Bc=0 ALIGN=MIDDLE  ALT=\"at%5E2%2Bbt%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%5E2%2B-69t%2B54+=+0 ALIGN=MIDDLE  ALT=\"16t%5E2%2B-69t%2B54+=+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=t%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"t%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=%28-69%29%5E2-4%2A16%2A54=1305 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-69%29%5E2-4%2A16%2A54=1305\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1305 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--69%2B-sqrt%28+1305+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--69%2B-sqrt%28+1305+%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=t%5B1%5D+=+%28-%28-69%29%2Bsqrt%28+1305+%29%29%2F2%5C16+=+3.28514949176178 ALIGN=MIDDLE  ALT=\"t%5B1%5D+=+%28-%28-69%29%2Bsqrt%28+1305+%29%29%2F2%5C16+=+3.28514949176178\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B2%5D+=+%28-%28-69%29-sqrt%28+1305+%29%29%2F2%5C16+=+1.02735050823822 ALIGN=MIDDLE  ALT=\"t%5B2%5D+=+%28-%28-69%29-sqrt%28+1305+%29%29%2F2%5C16+=+1.02735050823822\"   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=16t%5E2%2B-69t%2B54 ALIGN=MIDDLE  ALT=\"16t%5E2%2B-69t%2B54\"   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=16t%5E2%2B-69t%2B54+=+16%28t-3.28514949176178%29%2A%28t-1.02735050823822%29 ALIGN=MIDDLE  ALT=\"16t%5E2%2B-69t%2B54+=+16%28t-3.28514949176178%29%2A%28t-1.02735050823822%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.28514949176178, 1.02735050823822.\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%2B-69%2Ax%2B54+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+16%2Ax%5E2%2B-69%2Ax%2B54+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );