document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201880>Question 1201880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A steel ball is projected downward at an angle of 48° with the horizontal from the top of the cliff 480 m high. Its initial velocity is 43 m/s.<BR>\n" );
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document.write( "a. How long will it take before striking the ground?<BR>\n" );
document.write( "b. How far from the base of the building will it strike?<BR>\n" );
document.write( "c. With what velocity will it hit the ground?<BR>\n" );
document.write( "d. At what angle with horizontal will it strike? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836433 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=836433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A steel ball is projected downward at an angle of 48° with the horizontal from the top of the cliff 480 m high. Its initial velocity is 43 m/s.<BR>\n" );
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document.write( "Using 10 m/sec/sec for gravity:\r<BR>\n" );
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document.write( "a. How long will it take before striking the ground?<BR>\n" );
document.write( "The vertical component is -43*sin(48) = ~31.96 m/s<BR>\n" );
document.write( "h(t) = -5t^2 - 31.96t + 480 = 0 at impact<BR>\n" );
document.write( "Solve for t<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=-5x%5E2%2B-31.96x%2B480+=+0 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B-31.96x%2B480+=+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=%28-31.96%29%5E2-4%2A-5%2A480=10621.4416 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-31.96%29%5E2-4%2A-5%2A480=10621.4416\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=10621.4416 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--31.96%2B-sqrt%28+10621.4416+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--31.96%2B-sqrt%28+10621.4416+%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-%28-31.96%29%2Bsqrt%28+10621.4416+%29%29%2F2%5C-5+=+-13.5020378419643 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-31.96%29%2Bsqrt%28+10621.4416+%29%29%2F2%5C-5+=+-13.5020378419643\"   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-%28-31.96%29-sqrt%28+10621.4416+%29%29%2F2%5C-5+=+7.11003784196429 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-31.96%29-sqrt%28+10621.4416+%29%29%2F2%5C-5+=+7.11003784196429\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x%5E2%2B-31.96x%2B480 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B-31.96x%2B480\"   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=-5x%5E2%2B-31.96x%2B480+=+%28x--13.5020378419643%29%2A%28x-7.11003784196429%29 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B-31.96x%2B480+=+%28x--13.5020378419643%29%2A%28x-7.11003784196429%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -13.5020378419643, 7.11003784196429.\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+-5%2Ax%5E2%2B-31.96%2Ax%2B480+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-5%2Ax%5E2%2B-31.96%2Ax%2B480+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Ignore the negative value.<BR>\n" );
document.write( "t = 7.11 seconds to impact<BR>\n" );
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document.write( "b. How far from the base of the building will it strike?<BR>\n" );
document.write( "Horizontal component = 43*cos(48)<BR>\n" );
document.write( "Distance = 7.11*43*cos(48) = ~ 204.57 meters<BR>\n" );
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document.write( "c. With what velocity will it hit the ground?<BR>\n" );
document.write( "Impact speed = vert comp of 43 + 5t = 31.96 + 7.11*5 = ~ 67.51 m/sec<BR>\n" );
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document.write( "d. At what angle with horizontal will it strike?<BR>\n" );
document.write( "The path of the object is a parabola --- h(t) = -5t^2 - 31.96t + 480 = 0 at impact.<BR>\n" );
document.write( "Find the slope at t = 7.11<BR>\n" );
document.write( "h'(t) = -10t - 31.96 = -103.06<BR>\n" );
document.write( "Angle = 180 - h'(t) = 76.94 degs </SPAN>\n" );
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