document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194461>Question 194461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the function h=-5t2+20t+1 models the height, h meters, of a baseball as a function of the time, t seconds, since it was hit. The ball hit the ground before the fielder could catch it. Use the quadratic formula to solve the following problems.<BR>\n" );
document.write( "a) How long was the baseball in the air, to the nearest tenth of a second?<BR>\n" );
document.write( "b) For how many seconds was the height of the ball at least 16 m? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145906 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=145906\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the function h=-5t2+20t+1 models the height, h meters, of a baseball as a function of the time, t seconds, since it was hit. The ball hit the ground before the fielder could catch it. Use the quadratic formula to solve the following problems.<BR>\n" );
document.write( "a) How long was the baseball in the air, to the nearest tenth of a second?<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Set h=0 and solve for t<BR>\n" );
document.write( "h=-5t2+20t+1<BR>\n" );
document.write( "0=-5t2+20t+1<BR>\n" );
document.write( "Solving via the quadratic equation yields:<BR>\n" );
document.write( "x ={-0.04939, 4.0494}<BR>\n" );
document.write( "See below for details...<BR>\n" );
document.write( "We can toss out the negative solution leaving:<BR>\n" );
document.write( "x = 4.0494 seconds<BR>\n" );
document.write( ".<BR>\n" );
document.write( "b) For how many seconds was the height of the ball at least 16 m?<BR>\n" );
document.write( "Set h=0 and solve for t<BR>\n" );
document.write( "h=-5t2+20t+1<BR>\n" );
document.write( "16=-5t2+20t+1<BR>\n" );
document.write( "0=-5t2+20t-15<BR>\n" );
document.write( "0 = (-5t+15)(t-1)<BR>\n" );
document.write( "t = {1,3}<BR>\n" );
document.write( "This means on the way up at 1 second it reaches 16m and then on the way down at 3 seconds it passes 16m again.<BR>\n" );
document.write( "Thus it was in the air for 2 seconds.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Quadratic solution for part a:<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=quadratic.solver>SOLVE quadratic equation with variable</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%2B20x%2B1+=+0 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B20x%2B1+=+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=%2820%29%5E2-4%2A-5%2A1=420 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2820%29%5E2-4%2A-5%2A1=420\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=420 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+=+%28-20%2B-sqrt%28+420+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-20%2B-sqrt%28+420+%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-%2820%29%2Bsqrt%28+420+%29%29%2F2%5C-5+=+-0.0493901531919196 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2820%29%2Bsqrt%28+420+%29%29%2F2%5C-5+=+-0.0493901531919196\"   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-%2820%29-sqrt%28+420+%29%29%2F2%5C-5+=+4.04939015319192 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2820%29-sqrt%28+420+%29%29%2F2%5C-5+=+4.04939015319192\"   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=-5x%5E2%2B20x%2B1 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B20x%2B1\"   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%2B20x%2B1+=+-5%28x--0.0493901531919196%29%2A%28x-4.04939015319192%29 ALIGN=MIDDLE  ALT=\"-5x%5E2%2B20x%2B1+=+-5%28x--0.0493901531919196%29%2A%28x-4.04939015319192%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.0493901531919196, 4.04939015319192.\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%2B20%2Ax%2B1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-5%2Ax%5E2%2B20%2Ax%2B1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );