Question 122751
The correct solution to this problem is t = 4.912 seconds (rounded to 3 decimal places).
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The equation to use to solve this problem is:
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{{{h(t) = (-9.8/2)t^2 + 20t + 100}}}
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This equation gives the height as a function of time after the launch of the ball.
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The interpretation of the right side of this equation is that the first term accounts 
for the effects of gravity; the second term accounts for the initial velocity and the positive
sign that it has in the problem indicates that the ball is tossed upward from the building; and
the third term indicates that the building the ball is launched from is 100 meters tall 
(note that t = 0 at launch, and when t = 0 the first two terms on the right side become 0. 
Therefore the initial height as indicated by the right side is just 100).
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The problem tells you to find the time it takes for the ball to reach a height of 80 meters.
So for h(t) you can substitute 80 and the equation becomes:
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{{{80 = (-9.8/2)t^2 + 20t + 100}}}
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Subtract 80 from both sides of this equation to get rid of the 80 on the left side, and 
the equation becomes:
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{{{0 = (-9.8/2)t^2 + 20t + 20}}}
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Transpose this equation (switch sides) to get it into the standard quadratic form of:
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{{{(-9.8/2)t^2 + 20t + 20 = 0}}}
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This is in the standard form {{{at^2 + bt + c=0}}} and by comparing this standard form
with your equation you can see that a = -9.8/2 or -4.9, b = +20, and c = +20.
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The quadratic formula tells you that the solution equation for the standard form is:
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{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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So to get the solution for your problem all you have to do is to substitute the values you
have for a, b, and c into the solution equation and simplify it.
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Making the substitutions for a, b, and c in the solution equation results in:
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{{{t = (-(20) +- sqrt( 20^2-4*(-4.9)*20 ))/(2*(-4.9)) }}}
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The terms inside the radical are {{{20^2-4*(-4.9)*20}}} and they multiply out to  give:
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{{{20^2-4*(-4.9)*20 = 400 - (-392)=792}}}
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Substituting this for the numbers inside the radical results in the solution equation becoming:
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{{{t = (-(20) +- sqrt( 792))/(2*(-4.9)) }}}
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and {{{sqrt(792) = 28.14249456}}}
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Substituting this into the solution equation results in:
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{{{t = (-(20) +-28.14249456 )/(2*(-4.9)) }}}
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Multiplying out the denominator results in the equation becoming:
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{{{t = (-(20) +- 28.14249456)/(-9.8) }}}
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And removing the parentheses around the first term in the numerator (the 20) simplifies
the equation to:
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{{{t = (-20 +- 28.14249456)/(-9.8) }}}
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Notice that the answer needs to be positive because negative time is before the launch and
that doesn't make sense. Therefore, since the denominator is negative, the numerator also 
has to be negative to give a positive answer. This tells you that you can ignore the + sign
in the numerator because it will give a positive numerator. Therefore, the only answer that
will make sense comes from:
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{{{t = (-20 - 28.14249456)/(-9.8) }}}
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Combining the two terms in the numerator reduces the equation to:
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{{{t = ( - 48.14249456)/(-9.8) }}}
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and when you divide the numerator by the denominator you get an answer of:
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{{{t = 4.912499445}}}
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So the time for the object to be 80 meters above the ground is 4.912499445 seconds after
launch and this rounds off to 4.912 seconds.
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You can graph the flight path of the object by making the vertical axis the height and the
horizontal axis the time in seconds. Then graph {{{h(t) = -4.9t^2 + 20t + 100}}} and you get:
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{{{graph(500,500,-1,10,-5,150, -4.9x^2 + 20x + 100)}}}
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Notice on the graph that when t is zero, the height on the vertical axis is 100 meters, which
is the height from which the object is launched.
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Also notice that when t is approximately 4.912 seconds the height is approximately 80 
meters and the object will hit the ground in around 7 seconds.
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Hope this helps to get you on the right track with this problem and gives you some confidence
that this answer is correct. 
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