To determine the initial height of the ball and the time interval before the ball hits the ground from the given table, let's analyze the provided data step by step.
1. Initial Height of the Ball:
- Look at the value of [tex]\( h(t) \)[/tex] when [tex]\( t = 0 \)[/tex]. This value represents the initial height of the ball.
- From the table, when [tex]\( t = 0 \)[/tex], [tex]\( h(0) = 150 \)[/tex].
- Therefore, the initial height of the ball is [tex]\( 150 \)[/tex] units.
2. Time Interval before the Ball Hits the Ground:
- A ball hits the ground when its height becomes zero or less.
- Examine the values of [tex]\( h(t) \)[/tex] to find when the height changes from positive to negative or zero.
- From the table, we observe:
- [tex]\( h(5) = 27.5 \)[/tex]
- [tex]\( h(6) = -26.4 \)[/tex]
- The height of the ball changes from positive (27.5 units) at [tex]\( t = 5 \)[/tex] seconds to negative (-26.4 units) at [tex]\( t = 6 \)[/tex] seconds.
- This indicates that the ball hits the ground sometime between [tex]\( t = 5 \)[/tex] and [tex]\( t = 6 \)[/tex] seconds.
Based on this analysis, the correct initial height of the ball and the correct time interval before the ball hits the ground are:
- Initial height [tex]\( = 150 \)[/tex]
- Hits the ground between 5 and 6 seconds
Therefore, among the provided options, the correct choice is:
- Initial height [tex]\( = 150 \)[/tex]; hits the ground between 5 and 6 seconds.
