document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=23297>Question 23297</A>: <I><SPAN STYLE=\"word-break: break-all;\"> `SUPPOSE YOU THROW A BASEBALL STRAIGHT UP AT A VELOCITY OF 64 FEET PER SECOND.<BR>\n" );
document.write( "A FUNCTION<BR>\n" );
document.write( "CAN BE CREATED BY EXPRESSING DISTANCE ABOVE GROUND, S, AS A FUNCTION OF TIME, T.<BR>\n" );
document.write( "THIS FUNCTION IS: S=-16T^2+^V0^T+^S0\r<BR>\n" );
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document.write( "16 REPRESENTS 1/2G, THE GRAVITATIONAL PULL DUE TO GRAVITY (MEASURED IN FEET PER<BR>\n" );
document.write( "SECOND^2).<BR>\n" );
document.write( "^V0 IS THE INITIAL VELOCITY (HOW HARD DO YOU THROW THE OBJECT, MEASURED IN FEET<BR>\n" );
document.write( "PER SECOND). <BR>\n" );
document.write( "^S0 IS THE INITIAL DISTANCE ABOVE GROUND (IN FEET). IF YOU ARE STANDING ON THE<BR>\n" );
document.write( "GROUND, THEN ^S0=0.\r<BR>\n" );
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document.write( "How long will it take to hit the ground?<BR>\n" );
document.write( "What is the maximum height of the ball? Waht time was the maximum height attained? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #12085 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=12085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%28t%29+=+-16t%5E2+%2B+64t ALIGN=MIDDLE  ALT=\"s%28t%29+=+-16t%5E2+%2B+64t\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the time (t) at which the ball returns to the ground (s=0), set the above equation equalto 0 and solve for t.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16t%5E2%2B64t ALIGN=MIDDLE  ALT=\"0+=+-16t%5E2%2B64t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out a t.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+t%28-16t%2B64%29 ALIGN=MIDDLE  ALT=\"0+=+t%28-16t%2B64%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Apply the zero product principle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+0 ALIGN=MIDDLE  ALT=\"t+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and/or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-64t%2B64+=+0 ALIGN=MIDDLE  ALT=\"-64t%2B64+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The solution t=0 would be the situation at the start of this operation. So we are left with:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t%2B64+=+0 ALIGN=MIDDLE  ALT=\"-16t%2B64+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 64t to both sides of the equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64+=+16t ALIGN=MIDDLE  ALT=\"64+=+16t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 16.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+4 ALIGN=MIDDLE  ALT=\"t+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The ball returns to the ground at t = 4 seconds.\r<BR>\n" );
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document.write( "To find the maximum height reached by the ball you first will find the time, t, at which the function s(t) is a maximum.  Recall that the quadratic function represents a parabola and the maximum (or minimum) point on a parabola occurs at its vertex. The vertex, in this case, will be a maximum value because the parabola opens downward. How do you know this?...the coefficient of t^2 is negative.\r<BR>\n" );
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document.write( "The x-coordinate (or, in this case, the t-coordinate) of the vertex is given by:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-b%2F2a ALIGN=MIDDLE  ALT=\"t+=+-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a=-16 and b=64.  This is taken from the standard form for the quadratic equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=at%5E2%2Bbt%2Bc=0 ALIGN=MIDDLE  ALT=\"at%5E2%2Bbt%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The t-coordinate of the vertex is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-64%29%2F2%28-16%29 ALIGN=MIDDLE  ALT=\"t+=+%28-64%29%2F2%28-16%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+=+%28-64%29%2F%28-32%29 ALIGN=MIDDLE  ALT=\"t+=+%28-64%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+=+2 ALIGN=MIDDLE  ALT=\"t+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > The maximum height occurs at time t=2 seconds. Find the maximum height by substituting this value of t into the original function: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%28t%29+=+-16t%5E2%2B64t ALIGN=MIDDLE  ALT=\"s%28t%29+=+-16t%5E2%2B64t\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%282%29+=+-16%282%29%5E2+%2B+64%282%29 ALIGN=MIDDLE  ALT=\"s%282%29+=+-16%282%29%5E2+%2B+64%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%282%29+=+-64%2B128 ALIGN=MIDDLE  ALT=\"s%282%29+=+-64%2B128\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%282%29+=+64 ALIGN=MIDDLE  ALT=\"s%282%29+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The maximum height attained by the ball is 64 feet at 2 seconds. </SPAN>\n" );
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