The table relates to a function [tex]\( h(t) \)[/tex] that models the height of a ball [tex]\( t \)[/tex] seconds after it is dropped.
[tex]\[
\begin{tabular}{|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{tabular}
\][/tex]

Determine the initial height of the ball and the time interval before the ball hits the ground:

A. Initial height [tex]\( = 0 \)[/tex]; hits the ground between 5 and 6 seconds

B. Initial height [tex]\( = 150 \)[/tex]; hits the ground between 5 and 6 seconds

C. Initial height [tex]\( = 0 \)[/tex]; hits the ground between 3 and 4 seconds

D. Initial height [tex]\( = 150 \)[/tex]; hits the ground between 4 and 5 seconds



Answer :

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]