document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190294>Question 190294</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A signal flare is fired upward with initial speed 245m/s.  A stationary balloonist at the height of 1960 m sees the flare pass on the way up.  How long after this will the flare pass the balloonist again on the way down?\r<BR>\n" );
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document.write( "This is for my son, and yes, I'm a few years out of practice. He's tried setting it to zero with: -4.9t^2-245t+1960=0. I think h=rt-4.9t^2 is the formula was also given to him. Can you help, please? He knows the answer is 30 seconds, but can't show the work. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142812 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=142812\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The height function is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  h(t) = -4.9t^2 + 245t + h_o\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h_o\"> is the initial height of the projectile.  You could argue that if it were a 6' tall man firing the flare gun, then the initial height is about 2 meters, but I suspect that this little detail was ignored for this problem, so let's just set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h_o = 0\">.  That gives us a height function that looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  h(t) = -4.9t^2 + 245t \">\r<BR>\n" );
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document.write( "What this means is that you can calculate the instantaneous height for any value of <i>t</i>.  But that doesn't help directly because we are given the height and we need to discover the time.  Hence, the question becomes: \"What is the value of <i>t</i> when the height is 1960 meters\", or:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  -4.9t^2 + 245t = 1960\">\r<BR>\n" );
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document.write( "Put the quadratic into standard form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  -4.9t^2 + 245t - 1960 = 0\">\r<BR>\n" );
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document.write( "Solve with the quadratic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{-b \pm sqrt{b^2 - 4ac}}{2a} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{-245 \pm sqrt{245^2 - 4(-4.9)(-1960)}}{2(-4.9)} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{-245 \pm sqrt{21609}}{-9.8} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{-245 \pm 147}{-9.8} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t_1 = 10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t_2 = 40\">\r<BR>\n" );
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document.write( "So the flare fired at time 0 gets to the balloonist at time 10 seconds and then passes him on the way back down at 40 seconds, i.e. 30 seconds later.\r<BR>\n" );
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document.write( "<i><b>Extra credit:</b></i>  How many seconds after the flare passes the balloonist does it hit the ground?\r<BR>\n" );
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document.write( "<i><b>Super-double-plus extra credit:</b></i>  When does the flare get as high as it will ever get, and how high is that?\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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