document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81891>Question 81891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Everyone in my family has tried to answer this worksheet from school.  Does it need to be solved before graphing?  Would appreciate your help with this problem.  Thanks!  \r<BR>\n" );
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document.write( "A rocket is launched from atop a 78-foot cliff with an initial velocity of 152 feet per second.  the height of the rocket t seconds after launch is given by the equation h=-16t^2 + 152t + 78.  Graph the equation to find out how long after the rocket is launched it will hit the ground.  Estimate the answer to the nearest tenth of a second.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58692 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=58692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-16t%5E2+%2B+152t+%2B+78 ALIGN=MIDDLE  ALT=\"h+=+-16t%5E2+%2B+152t+%2B+78\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can solve this problem by graphing only.  Label the vertical axis (normally the y-axis)<BR>\n" );
document.write( "as height and label the horizontal axis (normally the x-axis) as time.  Then you can make<BR>\n" );
document.write( "the graph by picking values of t, substituting those values into the equation and finding<BR>\n" );
document.write( "the corresponding value of h, the height.<BR>\n" );
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document.write( "For example, when t is zero the height is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-16%280%5E2%29+%2B+152%280%29+%2B+78+=+0+%2B+0+%2B+78+=+78 ALIGN=MIDDLE  ALT=\"h+=+-16%280%5E2%29+%2B+152%280%29+%2B+78+=+0+%2B+0+%2B+78+=+78\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So when the time is zero, the height is 78 feet, but the problem told you that.<BR>\n" );
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document.write( "On the height (vertical axis) put a dot at 78 feet.  <BR>\n" );
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document.write( "Next try letting t = 1 second. The equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-16t%5E2+%2B+152t+%2B+78+=+-16%281%5E2%29+%2B+152%281%29+%2B+78+=+-16+%2B+152+%2B+78+=+214 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ft<BR>\n" );
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document.write( "Plot the point (1, 214). That's a second point on the graph. If you continue picking more<BR>\n" );
document.write( "values for t, you will begin to see a pattern in which the object rises to a peak, and then<BR>\n" );
document.write( "starts to drop back to Earth.  <BR>\n" );
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document.write( "It takes a lot of work to build a graph that is complete enough to get the time it hits<BR>\n" );
document.write( "the ground down to 0.1 second accuracy.<BR>\n" );
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document.write( "So let's do it mathematically. It will tell you where to concentrate your graph.  <BR>\n" );
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document.write( "Begin by recognizing that when the rocket hits the ground its height will be zero. So<BR>\n" );
document.write( "in the equation, we can set h equal to zero and solve for t.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16t%5E2+%2B+152t+%2B+78 ALIGN=MIDDLE  ALT=\"0+=+-16t%5E2+%2B+152t+%2B+78\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Normally by convention we write this with the sides switched. So ...<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t%5E2+%2B+152t+%2B+78+=+0 ALIGN=MIDDLE  ALT=\"-16t%5E2+%2B+152t+%2B+78+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is of the standard form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=at%5E2+%2B+b%2At+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"at%5E2+%2B+b%2At+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and by comparison with our equation we can see that a = -16, b = 152, and c = 78.<BR>\n" );
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document.write( "The quadratic formula tells us that for equations in the standard form, the solutions<BR>\n" );
document.write( "to the equation are given by the equation:<BR>\n" );
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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" );
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document.write( "So now all you have to do is substitute the values of a, b, and c and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-152+%2B-+sqrt%28152%5E2+-+4%2A%28-16%29%2A78%29%29%2F%282%2A%28-16%29%29 ALIGN=MIDDLE  ALT=\"t+=+%28-152+%2B-+sqrt%28152%5E2+-+4%2A%28-16%29%2A78%29%29%2F%282%2A%28-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now it's just a lot of calculator work. Note that 152^2 = 23104 and -4*(-16)*78 = 4992. Substitute <BR>\n" );
document.write( "and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-152+%2B-+sqrt%2823104+%2B+4992%29%29%2F%28-32%29+ ALIGN=MIDDLE  ALT=\"t+=+%28-152+%2B-+sqrt%2823104+%2B+4992%29%29%2F%28-32%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-152+%2B-+sqrt%2828096%29%29%2F%28-32%29+=+%28-152+%2B-+167.6186147%29%2F%28-32%29 ALIGN=MIDDLE  ALT=\"t+=+%28-152+%2B-+sqrt%2828096%29%29%2F%28-32%29+=+%28-152+%2B-+167.6186147%29%2F%28-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This will lead to two answers ... one with the plus sign and one with the minus sign.  Let's solve<BR>\n" );
document.write( "for the two possible answers. First solve using the plus sign. That leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-152+%2B+167.6186147%29%2F%28-32%29+=+15.61861472%2F%28-32%29+=+-+0.488081709 ALIGN=MIDDLE  ALT=\"t+=+%28-152+%2B+167.6186147%29%2F%28-32%29+=+15.61861472%2F%28-32%29+=+-+0.488081709\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But this cannot be ... a minus time means before launch. So we know this answer is wrong.<BR>\n" );
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document.write( "So let's solve for the other possible answer. This time use the minus sign:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-152+-+167.6186147%29%2F%28-32%29+=+-319.6186147%2F%28-32%29+=+9.98808171 ALIGN=MIDDLE  ALT=\"t+=+%28-152+-+167.6186147%29%2F%28-32%29+=+-319.6186147%2F%28-32%29+=+9.98808171\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells us that 9.98808171 seconds after launch the rocket hits the ground. Now you<BR>\n" );
document.write( "know what region of time you need to concentrate on. To the nearest tenth of a second the<BR>\n" );
document.write( "answer is 10.0 seconds).<BR>\n" );
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document.write( "And for further information, the highest the rocket will get in its flight occurs when<BR>\n" );
document.write( "the time is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds.  Substituting the values of \"b\" and \"a\" above the<BR>\n" );
document.write( "answer becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-152%2F%282%2A%28-16%29%29+=+-152%2F-32+=+4.75 ALIGN=MIDDLE  ALT=\"-152%2F%282%2A%28-16%29%29+=+-152%2F-32+=+4.75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When t = 4.75 seconds after launch, the rocket will be at its maximum height.  All you <BR>\n" );
document.write( "have to do to find the maximum height is to use the height equation and set t equal to <BR>\n" );
document.write( "4.75 seconds:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-16t%5E2+%2B+152t+%2B+78+=+-16%2A%284.75%5E2%29%2B152%2A%284.75%29+%2B+78+=+-361+%2B+722+%2B+78+=+439+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So in your graphing you know that the maximum height the rocket will reach is 439 feet<BR>\n" );
document.write( "and that will occur when the time is 4.75 seconds after launch.<BR>\n" );
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document.write( "Hope this helps you to understand and do the problem.  </SPAN>\n" );
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