document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=371009>Question 371009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Aubrey uses a slingshot to shoot a marble from a window. The height of the marble h(t) in feet at time t seconds is given by the function h(t)=-16t square -50t+164. How many seconds elapse before the marble hits 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 #264335 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=264335\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> h(t)=-16t square -50t+164<BR>\n" );
document.write( "When it reaches the ground the \"height\" is zero<BR>\n" );
document.write( "so, set h(t) to zero and solve for t:<BR>\n" );
document.write( "0=-16t^2 -50t+164<BR>\n" );
document.write( "0=16t^2 + 50t - 164<BR>\n" );
document.write( "0=8t^2 + 25t - 82<BR>\n" );
document.write( "Applying the \"quadratic formula\" we find that<BR>\n" );
document.write( "t = {2, -5.125}<BR>\n" );
document.write( "Throw out the negative answer leaving<BR>\n" );
document.write( "t = 2 seconds<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Details of quadratic to follow:<BR>\n" );
document.write( ".<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=8t%5E2%2B25t%2B-82+=+0 ALIGN=MIDDLE  ALT=\"8t%5E2%2B25t%2B-82+=+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=%2825%29%5E2-4%2A8%2A-82=3249 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2825%29%5E2-4%2A8%2A-82=3249\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=3249 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+=+%28-25%2B-sqrt%28+3249+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-25%2B-sqrt%28+3249+%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-%2825%29%2Bsqrt%28+3249+%29%29%2F2%5C8+=+2 ALIGN=MIDDLE  ALT=\"t%5B1%5D+=+%28-%2825%29%2Bsqrt%28+3249+%29%29%2F2%5C8+=+2\"   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-%2825%29-sqrt%28+3249+%29%29%2F2%5C8+=+-5.125 ALIGN=MIDDLE  ALT=\"t%5B2%5D+=+%28-%2825%29-sqrt%28+3249+%29%29%2F2%5C8+=+-5.125\"   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=8t%5E2%2B25t%2B-82 ALIGN=MIDDLE  ALT=\"8t%5E2%2B25t%2B-82\"   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=8t%5E2%2B25t%2B-82+=+8%28t-2%29%2A%28t--5.125%29 ALIGN=MIDDLE  ALT=\"8t%5E2%2B25t%2B-82+=+8%28t-2%29%2A%28t--5.125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 2, -5.125.\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%2B25%2Ax%2B-82+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B25%2Ax%2B-82+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );