To find out how long it takes for the ball to reach a height of 72 feet, we can set the height equation equal to 72 and solve for \( t \).
The height equation is given as:
\[ S = 12 + 64t - 16t^2 \]
We want to find \( t \) when \( S = 72 \):
\[ 72 = 12 + 64t - 16t^2 \]
Let's rearrange this equation:
\[ 72 - 12 = 64t - 16t^2 \]
\[ 60 = 64t - 16t^2 \]
Now let's move everything to one side of the equation:
\[ -16t^2 + 64t - 60 = 0 \]
To simplify this equation, we can multiply through by \(-1\):
\[ 16t^2 - 64t + 60 = 0 \]
Next, we can divide the entire equation by 4 to further simplify:
\[ 4t^2 - 16t + 15 = 0 \]
Now we can use the quadratic formula to solve for \( t \):
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4 \), \( b = -16 \), and \( c = 15 \). Plugging in these values:
\[ t = \frac{-(-16) \pm \sqrt{(-16)^2 - 4 \cdot 4 \cdot 15}}{2 \cdot 4} \]
\[ t = \frac{16 \pm \sqrt{256 - 240}}{8} \]
\[ t = \frac{16 \pm \sqrt{16}}{8} \]
\[ t = \frac{16 \pm 4}{8} \]
This gives us two possible solutions:
- \( t = \frac{20}{8} = \frac{5}{2} = 2.5 \)
- \( t = \frac{12}{8} = \frac{3}{2} = 1.5 \)
Thus, the times at which the ball reaches a height of 72 feet are:
- \( t = 1.5 \) seconds (when the ball is rising)
- \( t = 2.5 \) seconds (when the ball is falling)
So in answer to your question, it takes 1.5 seconds and 2.5 seconds for the ball to reach the height of 72 feet.
You can enter your response as:
1.5, 2.5
1 answer