document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=822621>Question 822621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During a baseball game, a player’s ball hit into the outfield is modeled by the equation h = –16t2 + 65t. A bird flying across the field follows the path modeled by the equation h = 8t + 20. Let t be the time since the ball is hit and h be the height. What do the intersection points of the equations represent? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495016 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495016\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ball's equation:<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h%28t%29+=+-16%2At%5E2+%2B+65t+ ALIGN=MIDDLE  ALT=\"+h%28t%29+=+-16%2At%5E2+%2B+65t+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Bird's equation:<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h%28t%29+=+8t+%2B+20+ ALIGN=MIDDLE  ALT=\"+h%28t%29+=+8t+%2B+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "To find the intersections, I'll say<BR>\n" );
document.write( "that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h%28t%29+ ALIGN=MIDDLE  ALT=\"+h%28t%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same for both<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-16%2At%5E2+%2B+65t++=+8t+%2B+20+ ALIGN=MIDDLE  ALT=\"+-16%2At%5E2+%2B+65t++=+8t+%2B+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-16t%5E2+%2B+57t+-+20+=+0+ ALIGN=MIDDLE  ALT=\"+-16t%5E2+%2B+57t+-+20+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+-16+ ALIGN=MIDDLE  ALT=\"+a+=+-16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+57+ ALIGN=MIDDLE  ALT=\"+b+=+57+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+-20+ ALIGN=MIDDLE  ALT=\"+c+=+-20+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-57+%2B-+sqrt%28+57%5E2+-+4%2A%28-16%29%2A%28-20%29+%29%29+%2F+%282%2A%28-16%29%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-57+%2B-+sqrt%28+57%5E2+-+4%2A%28-16%29%2A%28-20%29+%29%29+%2F+%282%2A%28-16%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-57+%2B-+sqrt%28+3249+-+1280+%29%29+%2F+%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-57+%2B-+sqrt%28+3249+-+1280+%29%29+%2F+%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-57+%2B-+sqrt%28+1969+%29%29+%2F+%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-57+%2B-+sqrt%28+1969+%29%29+%2F+%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-57+%2B+44.373%29+%2F+%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-57+%2B+44.373%29+%2F+%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28+-12.627+%29%2F%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28+-12.627+%29%2F%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+.395+ ALIGN=MIDDLE  ALT=\"+t+=+.395+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and, also:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28-57+-+44.373%29+%2F+%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28-57+-+44.373%29+%2F+%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+%28+-101.373+%29%2F%28-32%29+ ALIGN=MIDDLE  ALT=\"+t+=+%28+-101.373+%29%2F%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+3.168+ ALIGN=MIDDLE  ALT=\"+t+=+3.168+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "These are the times at which the heights of the<BR>\n" );
document.write( "bird and the ball are the same<BR>\n" );
document.write( "Here's the plot of both trajectories:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-2%2C+6%2C+-10%2C+80%2C+-16x%5E2+%2B+65x%2C+8x+%2B+20+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+400%2C+400%2C+-2%2C+6%2C+-10%2C+80%2C+-16x%5E2+%2B+65x%2C+8x+%2B+20+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "What I would say the intersections represent is that<BR>\n" );
document.write( "if the bird is 20 feet directly above the batter at the<BR>\n" );
document.write( "exact moment the ball is hit, the 1st intersection is <BR>\n" );
document.write( "the height and time of the collision. <BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "If the bird wasn't exactly overhead, but followed the same<BR>\n" );
document.write( "path, the 2nd intersection is the only height and time<BR>\n" );
document.write( "for a possible collision.<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );