document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036786>Question 1036786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The height of a projectile t seconds after being launched with an initial<BR>\n" );
document.write( "velocity of 96 ft/sec from an initial height of 200 ft above the ground is given by the<BR>\n" );
document.write( "following formula:<BR>\n" );
document.write( "s(t) = &#8722;16t^2 + 96t + 200\r<BR>\n" );
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document.write( "(a) At what moment in time is the height of the projectile the greatest?<BR>\n" );
document.write( "(b) At what moment in time does the projectile return to the ground?<BR>\n" );
document.write( "(c) The slope of the line through points (t0, s(t0)) and (t1, s(t1)) is the average velocity<BR>\n" );
document.write( "on the time interval [t0, t1]. What is the average velocity of the particle from the<BR>\n" );
document.write( "moment it was launched to the moment when it reaches its greatest height? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651560 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=651560\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The height of a projectile t seconds after being launched with an initial<BR>\n" );
document.write( "velocity of 96 ft/sec from an initial height of 200 ft above the ground is given by the<BR>\n" );
document.write( "following formula:<BR>\n" );
document.write( "s(t) = -16t^2 + 96t + 200\r<BR>\n" );
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document.write( "(a) At what moment in time is the height of the projectile the greatest?<BR>\n" );
document.write( "\"moment in time\" is redundant.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "(a) At what time is the height of the projectile the greatest?<BR>\n" );
document.write( "It's the vertex of the parabola, at t = -b/2a<BR>\n" );
document.write( "t = -96/-32 = 3 seconds.<BR>\n" );
document.write( "=========================<BR>\n" );
document.write( "(b) At what time does the projectile return to the ground?<BR>\n" );
document.write( "When s(t) = 0.<BR>\n" );
document.write( "-16t^2 + 96t + 200 = 0<BR>\n" );
document.write( "-2t^2 + 12t + 25 = 0<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=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</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=-2x%5E2%2B12x%2B25+=+0 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B12x%2B25+=+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=%2812%29%5E2-4%2A-2%2A25=344 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2812%29%5E2-4%2A-2%2A25=344\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=344 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+=+%28-12%2B-sqrt%28+344+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-12%2B-sqrt%28+344+%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-%2812%29%2Bsqrt%28+344+%29%29%2F2%5C-2+=+-1.63680924774785 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+344+%29%29%2F2%5C-2+=+-1.63680924774785\"   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-%2812%29-sqrt%28+344+%29%29%2F2%5C-2+=+7.63680924774785 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2812%29-sqrt%28+344+%29%29%2F2%5C-2+=+7.63680924774785\"   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=-2x%5E2%2B12x%2B25 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B12x%2B25\"   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=-2x%5E2%2B12x%2B25+=+%28x--1.63680924774785%29%2A%28x-7.63680924774785%29 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B12x%2B25+=+%28x--1.63680924774785%29%2A%28x-7.63680924774785%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -1.63680924774785, 7.63680924774785.\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+-2%2Ax%5E2%2B12%2Ax%2B25+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-2%2Ax%5E2%2B12%2Ax%2B25+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "======<BR>\n" );
document.write( "Ignore the negative solution.<BR>\n" );
document.write( "t =~ 7.637 seconds<BR>\n" );
document.write( "===============<BR>\n" );
document.write( "(c) The slope of the line through points (t0, s(t0)) and (t1, s(t1)) is the average velocity<BR>\n" );
document.write( "on the time interval [t0, t1]. What is the average velocity of the particle from the<BR>\n" );
document.write( "moment it was launched to the moment when it reaches its greatest height?<BR>\n" );
document.write( "--<BR>\n" );
document.write( "t0 = 0, t1 = 3<BR>\n" );
document.write( "s0 = 200, s1 = s(3) = -16*9 + 96*3 + 200 = 344<BR>\n" );
document.write( "--> (344-200)/3 = 48 ft/sec </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );