Answer :
Ok, let's analyze each statement one by one, based on the provided table of values for the height of the ball at different times:
[tex]\[ \begin{tabular}{|c|c|} \hline $t$ & $h ( t )$ \\ \hline 0 & 0 \\ \hline 2 & 60.4 \\ \hline 4 & 81.6 \\ \hline 6 & 63.6 \\ \hline 8 & 6.4 \\ \hline 10 & -90 \\ \hline 12 & -225.6 \\ \hline \end{tabular} \][/tex]
1. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- To check this, we need to see if the height at 8 seconds and 10 seconds is the same.
- At [tex]$t = 8$[/tex] seconds: [tex]$h(8) = 6.4$[/tex] meters.
- At [tex]$t = 10$[/tex] seconds: [tex]$h(10) = -90$[/tex] meters.
The heights at 8 seconds and 10 seconds are different, so this statement is false.
2. The height of the ball decreases and then increases.
- We need to observe the height values to see if there's a pattern where the height decreases and then increases.
- The height values increase from [tex]$0$[/tex] to [tex]$81.6$[/tex] at [tex]$t = 4$[/tex], then they decrease afterward.
The height increases initially and then decreases, so this statement is false.
3. The ball reaches its maximum height about 4 seconds after it is thrown.
- To check this, we need to determine the maximum height and its corresponding time.
- The maximum height from the table is [tex]$81.6$[/tex] meters at [tex]$t = 4$[/tex] seconds.
Therefore, the statement is true.
4. The ball hits the ground between 8 and 10 seconds after it is thrown.
- To determine this, we need to check if the height becomes zero or negative between these intervals.
- At [tex]$t = 8$[/tex] seconds: [tex]$h(8) = 6.4$[/tex] meters (still above the ground).
- At [tex]$t = 10$[/tex] seconds: [tex]$h(10) = -90$[/tex] meters (below the ground, implying it crossed zero before this time).
Since the ball goes below ground level between 8 and 10 seconds, this statement is true.
5. The height of the building is 81.6 meters.
- To determine the height of the building, we need to identify the height at [tex]$t = 0$[/tex] seconds or when the ball was thrown, which is 0 meters.
- However, if the maximum height reached by the ball is stated as the height of the building, it would be 81.6 meters.
The height of the building is therefore possibly misinterpreted as the maximum height attained by the ball in this context. Hence, this statement is true if we consider it aligns with the maximum height reached by the ball.
Thus, the results are as follows:
1. False
2. False
3. True
4. True
5. True
[tex]\[ \begin{tabular}{|c|c|} \hline $t$ & $h ( t )$ \\ \hline 0 & 0 \\ \hline 2 & 60.4 \\ \hline 4 & 81.6 \\ \hline 6 & 63.6 \\ \hline 8 & 6.4 \\ \hline 10 & -90 \\ \hline 12 & -225.6 \\ \hline \end{tabular} \][/tex]
1. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- To check this, we need to see if the height at 8 seconds and 10 seconds is the same.
- At [tex]$t = 8$[/tex] seconds: [tex]$h(8) = 6.4$[/tex] meters.
- At [tex]$t = 10$[/tex] seconds: [tex]$h(10) = -90$[/tex] meters.
The heights at 8 seconds and 10 seconds are different, so this statement is false.
2. The height of the ball decreases and then increases.
- We need to observe the height values to see if there's a pattern where the height decreases and then increases.
- The height values increase from [tex]$0$[/tex] to [tex]$81.6$[/tex] at [tex]$t = 4$[/tex], then they decrease afterward.
The height increases initially and then decreases, so this statement is false.
3. The ball reaches its maximum height about 4 seconds after it is thrown.
- To check this, we need to determine the maximum height and its corresponding time.
- The maximum height from the table is [tex]$81.6$[/tex] meters at [tex]$t = 4$[/tex] seconds.
Therefore, the statement is true.
4. The ball hits the ground between 8 and 10 seconds after it is thrown.
- To determine this, we need to check if the height becomes zero or negative between these intervals.
- At [tex]$t = 8$[/tex] seconds: [tex]$h(8) = 6.4$[/tex] meters (still above the ground).
- At [tex]$t = 10$[/tex] seconds: [tex]$h(10) = -90$[/tex] meters (below the ground, implying it crossed zero before this time).
Since the ball goes below ground level between 8 and 10 seconds, this statement is true.
5. The height of the building is 81.6 meters.
- To determine the height of the building, we need to identify the height at [tex]$t = 0$[/tex] seconds or when the ball was thrown, which is 0 meters.
- However, if the maximum height reached by the ball is stated as the height of the building, it would be 81.6 meters.
The height of the building is therefore possibly misinterpreted as the maximum height attained by the ball in this context. Hence, this statement is true if we consider it aligns with the maximum height reached by the ball.
Thus, the results are as follows:
1. False
2. False
3. True
4. True
5. True
