To address the problem, we begin by understanding the function that describes the height of the football, which is given as [tex]\(h(t) = -16t^2 + 64t\)[/tex]. This is a quadratic function of the form [tex]\(at^2 + bt + c\)[/tex], where [tex]\(a = -16\)[/tex] and [tex]\(b = 64\)[/tex].
### Finding the Axis of Symmetry
1. Formula for the Axis of Symmetry: The axis of symmetry for any quadratic equation [tex]\(ax^2 + bx + c\)[/tex] is found using the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
2. Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the Formula:
[tex]\[
t = -\frac{64}{2(-16)}
\][/tex]
3. Simplify the Calculation:
[tex]\[
t = -\frac{64}{-32} = 2
\][/tex]
So, the axis of symmetry, which represents the time at which the ball reaches its maximum height, is at [tex]\(t = 2\)[/tex] seconds.
### Relating the Axis of Symmetry to the Time the Ball is in the Air
- The axis of symmetry [tex]\(t = 2\)[/tex] seconds indicates it takes 2 seconds for the ball to reach its maximum height.
- Since the trajectory of the ball is symmetric around this point of time (i.e., the ball will take the same amount of time to come back down to the ground as it did to rise to its highest point), the total time in the air is twice this duration.
Thus, the total time the ball is in the air is [tex]\(2 \times 2\)[/tex] seconds = 4 seconds.
### Conclusion
- At [tex]\(t = 2\)[/tex], it takes the ball 2 seconds to reach its maximum height.
- The ball takes an additional 2 seconds to fall back to the ground.
- Therefore, the total time the ball is in the air is 4 seconds.
Hence, the correct statement is:
[tex]\(t = 2\)[/tex]; It takes the ball 2 seconds to reach the maximum height and 2 more seconds to fall back to the ground.
