document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=210045>Question 210045</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rocket is launched from atop a 43-foot cliff with an initial velocity of 70 feet per second. The height of the rocket t seconds after launch is given by the equation h = -16t^2 + 70t + 43. Graph the equation to find out how long after the rocket is launched it ill hit the ground. Estimate your answer to the nearest tenth of a second. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #158752 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=158752\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A rocket is launched from atop a 43-foot cliff with an initial velocity of 70 feet per second. The height of the rocket t seconds after launch is given by the equation h = -16t^2 + 70t + 43. Graph the equation to find out how long after the rocket is launched it ill hit the ground. Estimate your answer to the nearest tenth of a second.<BR>\n" );
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document.write( "h = -16t^2 + 70t + 43<BR>\n" );
document.write( "Set h = 0 and solve for t<BR>\n" );
document.write( "-16t^2 + 70t + 43 = 0<BR>\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=-16x%5E2%2B70x%2B43+=+0 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B70x%2B43+=+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=%2870%29%5E2-4%2A-16%2A43=7652 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2870%29%5E2-4%2A-16%2A43=7652\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=7652 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+=+%28-70%2B-sqrt%28+7652+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-70%2B-sqrt%28+7652+%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-%2870%29%2Bsqrt%28+7652+%29%29%2F2%5C-16+=+-0.54611596607863 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2870%29%2Bsqrt%28+7652+%29%29%2F2%5C-16+=+-0.54611596607863\"   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-%2870%29-sqrt%28+7652+%29%29%2F2%5C-16+=+4.92111596607863 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2870%29-sqrt%28+7652+%29%29%2F2%5C-16+=+4.92111596607863\"   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=-16x%5E2%2B70x%2B43 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B70x%2B43\"   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=-16x%5E2%2B70x%2B43+=+%28x--0.54611596607863%29%2A%28x-4.92111596607863%29 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B70x%2B43+=+%28x--0.54611596607863%29%2A%28x-4.92111596607863%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.54611596607863, 4.92111596607863.\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%2B70%2Ax%2B43+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-16%2Ax%5E2%2B70%2Ax%2B43+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Use the positive number.<BR>\n" );
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document.write( "BTW, rockets have motors and they accelerate when launched.  This is a ballistics problem, and the \"rocket\" just rises and falls due to initial forces and gravity.<BR>\n" );
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document.write( "For a better graph, get the FREE software from<BR>\n" );
document.write( "http://www.padowan.dk.com<BR>\n" );
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