document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=720461>Question 720461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We throw an object upward from the top I a 1200 ft. Building. The height of the object, (measured in feet) t seconds after we threw it is h = -16t^2+160t+1200<BR>\n" );
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document.write( "A) when will the object reach its highest point?\r<BR>\n" );
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document.write( "B) How long does it take for the object to bit the ground? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #441909 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=441909\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First we should recognize the the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"h+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be a parabola. Not only that, the parabola will open downward because of the negative coefficient, -16, in front of the squared term. If you picture (or draw) such a parabola, you should notice that the highest point on the graph is the vertex of this parabola. So we will find out when the object reaches the highest point by finding the t-coordinate of the vertex.<br><BR>\n" );
document.write( "One way to find the vertex of a parabola is to complete the square and transform the equation into vertex form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-h%29%5E2+=+4p%28H-k%29%0D%0Awhere%3Cul%3E%3Cli%3Et+is+the+number+of+seconds+since+the+object+was+thrown%3C%2Fli%3E%3Cli%3Eh+and+k+are+the+coordinates+of+the+vertex+%28h%2C+k%29%3C%2Fli%3E%3Cli%3EH+is+the+height+of+the+object+after+t+seconds.+%28I%27m+using+H+because+it+is+natural+to+name+a+height+as+h+but+the+ordered+pair+of+the+vertex+are+is+always+called+%28h%2C+k%29+so+we+would+have+two+h%27s.+To+try+to+avoid+confusion+I%27m+using+capital+H+for+the+height+and+h+for+the+coordinate+of+the+vertex.%3C%2Fli%3E%3C%2Ful%3EBut+there+is+a+much+quicker+way+if+you+remember+that+the+x-coordinate+%28in+this+case+t-coordinate%29+of+the+vertex+of+a+parabola+in+standard+%7B%7B%7Bax%5E2%2Bbx%2Bc ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > form is -b/2a. The b in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 160 and the a is -16. This makes the t-coordinate of the vertex:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-%28160%29%2F2%28-16%29+=+-160%2F%28-32%29+=+5 ALIGN=MIDDLE  ALT=\"t+=+-%28160%29%2F2%28-16%29+=+-160%2F%28-32%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the object will reach its highest point after 5 seconds. (If you need to find out how high this highest point is you would substitute in a 5 for the t into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for H.)<br><BR>\n" );
document.write( "The ground is at height zero. So to find out how long it will be before the ball comes back to the ground, we replace the H in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a 0 and solve for t:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"0+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring. First the greatest common factor (GCF), The GCF here is 16. But for reasons I will explain shortly, I cam going to factor out -16:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16%28t%5E2-10t-75%29 ALIGN=MIDDLE  ALT=\"0+=+-16%28t%5E2-10t-75%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "After factoring out the -16, the other factor has a squared term has a positive coefficient. This will make the rest of the factoring easier. This is why I factored out -16. The trinomial factors easily once we figure out that 5*15 = 75:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16%28t-15%29%28t%2B5%29 ALIGN=MIDDLE  ALT=\"0+=+-16%28t-15%29%28t%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we use the Zero Product Property:<BR>\n" );
document.write( "-16 = 0 or t-15 = 0 or t+5 = 0<BR>\n" );
document.write( "The first equation is a false statement. So we will not get a solution from that equation. But we can solve the other two equations:<BR>\n" );
document.write( "t = 15 or t = -5<BR>\n" );
document.write( "Since t is the number of seconds after the object was thrown, it makes no sense to have a negative time. (-5 would mean 5 seconds <i>before</i> the object was thrown!) So we reject that solution. This makes t = 15 the solution. After 15 seconds the object will hit the ground.<br><BR>\n" );
document.write( "P.S. In reality, if the object was thrown straight upwards from the top of the building, it would hit the top of the building on the way down. If the building's roof was flat the object would never reach the ground. It would get stuck on the roof. If the roof was slanted so that the object would bounce off and hit the ground, the equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2%2B160t%2B1200 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2%2B160t%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" >, would no longer give correct values after the bounce. </SPAN>\n" );
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