Let's analyze each statement based on the given data about the height of the ball over time:
[tex]\[
\begin{array}{|c|c|}
\hline
t & h(t) \\
\hline
0 & 0 \\
\hline
1 & 64 \\
\hline
2 & 96 \\
\hline
3 & 96 \\
\hline
4 & 64 \\
\hline
5 & 0 \\
\hline
\end{array}
\][/tex]
1. The initial height of the ball is 96 feet.
To check this statement, we look at the height of the ball at [tex]\( t = 0 \)[/tex]. According to the table, [tex]\( h(0) = 0 \)[/tex]. Therefore, the initial height of the ball is not 96 feet. This statement is false.
2. The ball will hit the ground between 2 and 3 seconds after it was thrown.
To check this statement, we need to observe the heights at times [tex]\( t = 2 \)[/tex] and [tex]\( t = 3 \)[/tex]. From the table, [tex]\( h(2) = 96 \)[/tex] and [tex]\( h(3) = 96 \)[/tex]. The ball is still in the air at both these times. However, we should consider the time interval immediately after 2 seconds up to 3 seconds. We should look at the next time instance, [tex]\( t = 4 \)[/tex] and [tex]\( t = 5\)[/tex]:
- [tex]\( h(4) = 64 \)[/tex]
- [tex]\( h(5) = 0 \)[/tex]
The ball hits the ground at [tex]\( t = 5 \)[/tex] seconds. But it doesn't hit the ground between 2 and 3 seconds. Hence, this statement is also false.
3. The maximum height of the ball must be 96 feet.
To check this statement, we need to find the maximum value in the height column of the table. The heights are: [tex]\( 0, 64, 96, 96, 64, 0 \)[/tex]. The maximum value in this list is 96 feet. This statement is correct.
4. The maximum height of the ball was reached 2.5 seconds after it was thrown.
To verify this statement, we need to find the exact time when the maximum height was reached. According to the table, the height 96 feet is reached at both [tex]\( t = 2 \)[/tex] and [tex]\( t = 3 \)[/tex] seconds. But there is no data given for t = 2.5 seconds explicitly. From the given data points (where peaks occur at t=2 and t=3), it would naturally imply the highest point was reached somewhere between 2 and 3 seconds, possibly at 2.5 seconds. Hence, we can consider this statement to be correct.
So, based on the given data and analysis, the following statements are true:
1. The maximum height of the ball must be 96 feet.
2. The maximum height of the ball was reached 2.5 seconds after it was thrown.
