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The equation y = -16t^2 - 60t + 54$ describes the height (in feet) of a ball thrown downward at 60 feet per second from a height of 54 feet from the ground. In how many seconds will the ball hit the ground? Express your answer as a decimal rounded to the nearest hundredth.

 May 16, 2019
 #1
avatar+8852 
+4

height  =  -16t2  - 60t + 54     , where  t  is the number of seconds after the ball is released

 

In how many seconds will the ball hit the ground?

In other words, what is  t  when the height is zero?

 

Plug in  0  for the height and solve for  t .

 

0  =  -16t2  - 60t + 54

                                             We can divide both sides of the equation by  -2

0  =  8t2 + 30t - 27

                                             Let's split  30t  into two terms such that their coefficients multiply to  -216

0  =  8t2 - 6t + 36t - 27

                                             Factor  2t  out of the first two terms and factor  9  out of the last two terms

0  =  2t(4t - 3) + 9(4t - 3)

                                             Factor  (4t - 3)  out of both remaining terms

0  =  (4t - 3)(2t + 9)

                                             Set each factor equal to  0  and solve for  t

4t - 3  =  0

_______or_______

2t + 9  =  0

 

 

4t  =  3

 

2t  =  -9  
t  =  3/4   t  =  -9/2

 

 

t  =  0.75   t  =  -4.5  

 

We want the number of seconds after the ball is released, so we only want the positive solution.

 

The ball will hit the ground  0.75  seconds.

 

Here's a graph: https://www.desmos.com/calculator/kznk4jayc4

 May 16, 2019
 #2
avatar+25 
-5

Can you use the quadratic formula to solve this?

ProffesorNobody  May 16, 2019
 #3
avatar+8852 
+3

Yep! You can use the quadratic formula to solve   -16t2  - 60t + 54  =  0   for  t , like this:

 

\(t \,=\, {-(-60) \,\pm \,\sqrt{(-60)^2-4(-16)(54)} \over 2(-16)}\,=\,{60\, \pm \,\sqrt{7056} \over -32}\,=\, {60 \,\pm\, 84 \over -32}\\~\\ \ \\ \begin{array}\ t\,=\,\frac{60+84}{-32}&\qquad\text{or}\qquad& t\,=\,\frac{60-84}{-32}\\~\\ t\,=\,\frac{144}{-32}&\qquad\text{or}\qquad& t\,=\,\frac{-24}{-32}\\~\\ t\,=\,-4.5&\qquad\text{or}\qquad& t\,=\,0.75 \end{array}\)

 

smiley

hectictar  May 16, 2019
edited by hectictar  May 16, 2019

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