To determine the initial height of the ball and the time interval before the ball hits the ground, let's analyze the given data in the table systematically.
First, note the values of [tex]\( h(t) \)[/tex], which represent the height of the ball at different time intervals [tex]\( t \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
t & h(t) \\
\hline
0 & 150 \\
\hline
1 & 145.1 \\
\hline
2 & 130.4 \\
\hline
3 & 105.9 \\
\hline
4 & 71.6 \\
\hline
5 & 27.5 \\
\hline
6 & -26.4 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Initial Height Determination:
- The initial height of the ball is the height at time [tex]\( t = 0 \)[/tex].
- From the table, when [tex]\( t = 0 \)[/tex], [tex]\( h(0) = 150 \)[/tex].
Therefore, the initial height of the ball is [tex]\( 150 \, \text{meters} \)[/tex].
2. Time Interval Before the Ball Hits the Ground:
- The ball hits the ground when the height [tex]\( h(t) \)[/tex] becomes non-positive (i.e., [tex]\( \leq 0 \)[/tex]).
- According to the table, the values of [tex]\( h(t) \)[/tex] at different times are:
- [tex]\( h(5) = 27.5 \)[/tex] (positive)
- [tex]\( h(6) = -26.4 \)[/tex] (negative)
Since [tex]\( h(t) \)[/tex] transitions from positive at [tex]\( t = 5 \)[/tex] to negative at [tex]\( t = 6 \)[/tex], it is clear that the ball hits the ground sometime between [tex]\( t = 5 \, \text{seconds} \)[/tex] and [tex]\( t = 6 \, \text{seconds} \)[/tex].
Thus, based on the given data:
- The initial height of the ball is [tex]\( 150 \, \text{meters} \)[/tex].
- The ball hits the ground in the time interval between 5 and 6 seconds.
### Conclusion:
Consequently, the correct answer is:
Initial height = 150; hits the ground between 5 and 6 seconds
Therefore, the correct choice is:
[tex]\[
\text{initial height } = 150; \text{ hits the ground between 5 and 6 seconds}
\][/tex]
