To solve this problem, we need to determine two things:
1. The initial height of the ball.
2. The time interval during which the ball hits the ground.
Let’s go through the solution step-by-step:
### Step 1: Determining the Initial Height
The initial height of the ball is given by the height at [tex]\( t = 0 \)[/tex]. From the table:
[tex]\[
\begin{array}{|c|c|}
\hline
t & h(t) \\
\hline
0 & 150 \\
\end{array}
\][/tex]
Therefore, the initial height of the ball is [tex]\( 150 \)[/tex] units.
### Step 2: Finding When the Ball Hits the Ground
The ball hits the ground when its height becomes zero or negative. We need to check values from the table to identify the time interval where the height transitions from positive to zero or negative.
[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]
At [tex]\( t = 5 \)[/tex], the height [tex]\( h(t) \)[/tex] is 27.5, which is positive. At [tex]\( t = 6 \)[/tex], [tex]\( h(t) \)[/tex] is -26.4, which is negative.
Since the ball's height transitions from positive at [tex]\( t = 5 \)[/tex] to negative at [tex]\( t = 6 \)[/tex], the ball must have hit the ground somewhere between [tex]\( t = 5 \)[/tex] and [tex]\( t = 6 \)[/tex].
### Conclusion
From our analysis:
- The initial height of the ball is [tex]\( 150 \)[/tex].
- The ball hits the ground between [tex]\( t = 5 \)[/tex] and [tex]\( t = 6 \)[/tex].
Thus, the correct answer is:
[tex]\[
\text{initial height } = 150; \text{ hits the ground between 5 and 6 seconds}
\][/tex]
