document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122751>Question 122751</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve.\r<BR>\n" );
document.write( "\n" );
document.write( "a ball is thrown upward from the roof of a building 100 m tall with an initial velocity of 20 m/s. When will the ball reach a height of 80 m? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #90154 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=90154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The correct solution to this problem is t = 4.912 seconds (rounded to 3 decimal places).<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The equation to use to solve this problem is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+100 ALIGN=MIDDLE  ALT=\"h%28t%29+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This equation gives the height as a function of time after the launch of the ball.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The interpretation of the right side of this equation is that the first term accounts <BR>\n" );
document.write( "for the effects of gravity; the second term accounts for the initial velocity and the positive<BR>\n" );
document.write( "sign that it has in the problem indicates that the ball is tossed upward from the building; and<BR>\n" );
document.write( "the third term indicates that the building the ball is launched from is 100 meters tall <BR>\n" );
document.write( "(note that t = 0 at launch, and when t = 0 the first two terms on the right side become 0. <BR>\n" );
document.write( "Therefore the initial height as indicated by the right side is just 100).<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The problem tells you to find the time it takes for the ball to reach a height of 80 meters.<BR>\n" );
document.write( "So for h(t) you can substitute 80 and the equation becomes:<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+100 ALIGN=MIDDLE  ALT=\"80+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract 80 from both sides of this equation to get rid of the 80 on the left side, and <BR>\n" );
document.write( "the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+20 ALIGN=MIDDLE  ALT=\"0+=+%28-9.8%2F2%29t%5E2+%2B+20t+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Transpose this equation (switch sides) to get it into the standard quadratic form of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9.8%2F2%29t%5E2+%2B+20t+%2B+20+=+0 ALIGN=MIDDLE  ALT=\"%28-9.8%2F2%29t%5E2+%2B+20t+%2B+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This is in the standard form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=at%5E2+%2B+bt+%2B+c=0 ALIGN=MIDDLE  ALT=\"at%5E2+%2B+bt+%2B+c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and by comparing this standard form<BR>\n" );
document.write( "with your equation you can see that a = -9.8/2 or -4.9, b = +20, and c = +20.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The quadratic formula tells you that the solution equation for the standard form is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So to get the solution for your problem all you have to do is to substitute the values you<BR>\n" );
document.write( "have for a, b, and c into the solution equation and simplify it.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Making the substitutions for a, b, and c in the solution equation results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-%2820%29+%2B-+sqrt%28+20%5E2-4%2A%28-4.9%29%2A20+%29%29%2F%282%2A%28-4.9%29%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-%2820%29+%2B-+sqrt%28+20%5E2-4%2A%28-4.9%29%2A20+%29%29%2F%282%2A%28-4.9%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The terms inside the radical are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%5E2-4%2A%28-4.9%29%2A20 ALIGN=MIDDLE  ALT=\"20%5E2-4%2A%28-4.9%29%2A20\"   DISABLED_style= \"max-width:90%; height:auto;\" > and they multiply out to  give:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%5E2-4%2A%28-4.9%29%2A20+=+400+-+%28-392%29=792 ALIGN=MIDDLE  ALT=\"20%5E2-4%2A%28-4.9%29%2A20+=+400+-+%28-392%29=792\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substituting this for the numbers inside the radical results in the solution equation becoming:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-%2820%29+%2B-+sqrt%28+792%29%29%2F%282%2A%28-4.9%29%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-%2820%29+%2B-+sqrt%28+792%29%29%2F%282%2A%28-4.9%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28792%29+=+28.14249456 ALIGN=MIDDLE  ALT=\"sqrt%28792%29+=+28.14249456\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substituting this into the solution equation results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-%2820%29+%2B-28.14249456+%29%2F%282%2A%28-4.9%29%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-%2820%29+%2B-28.14249456+%29%2F%282%2A%28-4.9%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Multiplying out the denominator results in the equation becoming:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-%2820%29+%2B-+28.14249456%29%2F%28-9.8%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-%2820%29+%2B-+28.14249456%29%2F%28-9.8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And removing the parentheses around the first term in the numerator (the 20) simplifies<BR>\n" );
document.write( "the equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-20+%2B-+28.14249456%29%2F%28-9.8%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-20+%2B-+28.14249456%29%2F%28-9.8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice that the answer needs to be positive because negative time is before the launch and<BR>\n" );
document.write( "that doesn't make sense. Therefore, since the denominator is negative, the numerator also <BR>\n" );
document.write( "has to be negative to give a positive answer. This tells you that you can ignore the + sign<BR>\n" );
document.write( "in the numerator because it will give a positive numerator. Therefore, the only answer that<BR>\n" );
document.write( "will make sense comes from:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-20+-+28.14249456%29%2F%28-9.8%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-20+-+28.14249456%29%2F%28-9.8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Combining the two terms in the numerator reduces the equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28+-+48.14249456%29%2F%28-9.8%29+ ALIGN=MIDDLE  ALT=\"t+=+%28+-+48.14249456%29%2F%28-9.8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and when you divide the numerator by the denominator you get an answer of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+4.912499445 ALIGN=MIDDLE  ALT=\"t+=+4.912499445\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the time for the object to be 80 meters above the ground is 4.912499445 seconds after<BR>\n" );
document.write( "launch and this rounds off to 4.912 seconds.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You can graph the flight path of the object by making the vertical axis the height and the<BR>\n" );
document.write( "horizontal axis the time in seconds. Then graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+-4.9t%5E2+%2B+20t+%2B+100 ALIGN=MIDDLE  ALT=\"h%28t%29+=+-4.9t%5E2+%2B+20t+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-1%2C10%2C-5%2C150%2C+-4.9x%5E2+%2B+20x+%2B+100%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-1%2C10%2C-5%2C150%2C+-4.9x%5E2+%2B+20x+%2B+100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice on the graph that when t is zero, the height on the vertical axis is 100 meters, which<BR>\n" );
document.write( "is the height from which the object is launched.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Also notice that when t is approximately 4.912 seconds the height is approximately 80 <BR>\n" );
document.write( "meters and the object will hit the ground in around 7 seconds.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps to get you on the right track with this problem and gives you some confidence<BR>\n" );
document.write( "that this answer is correct. <BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );