From the given table, we have the following data about the height of the ball at different times [tex]\( t \)[/tex] seconds after it is dropped:
[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]
1. Initial Height:
The initial height of the ball is the height at [tex]\( t = 0 \)[/tex]. From the table, when [tex]\( t = 0 \)[/tex], [tex]\( h(0) = 150 \)[/tex]. Thus, the initial height of the ball is 150.
2. Time Interval Before the Ball Hits the Ground:
The ball hits the ground when its height becomes zero or negative. From the table, we see that at [tex]\( t = 5 \)[/tex] seconds, [tex]\( h(5) \)[/tex] is 27.5, and at [tex]\( t = 6 \)[/tex] seconds, [tex]\( h(6) \)[/tex] is -26.4. The ball transitions from a positive height to a negative height between [tex]\( t = 5 \)[/tex] and [tex]\( t = 6 \)[/tex] seconds. Therefore, the ball hits the ground sometime between 5 and 6 seconds.
The correct interpretation from the table values is:
- Initial height [tex]\( = 150 \)[/tex]
- Hits the ground between 5 and 6 seconds
So the correct option is:
[tex]\[
\text{initial height } = 150; \text{ hits the ground between 5 and 6 seconds}
\][/tex]
