Answer :
To answer the given question, we need to analyze each statement according to the provided height data for different time points after the ball is thrown:
### Data Analysis:
The height values are given as follows:
- at [tex]\( t = 0\, \text{seconds}: h(t) = 0\)[/tex]
- at [tex]\( t = 2\, \text{seconds}: h(t) = 60.4\)[/tex]
- at [tex]\( t = 4\, \text{seconds}: h(t) = 81.6\)[/tex]
- at [tex]\( t = 6\, \text{seconds}: h(t) = 63.6\)[/tex]
- at [tex]\( t = 8\, \text{seconds}: h(t) = 6.4\)[/tex]
- at [tex]\( t = 10\, \text{seconds}: h(t) = -90\)[/tex]
- at [tex]\( t = 12\, \text{seconds}: h(t) = -225.6\)[/tex]
### Evaluation of Statements:
1. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- At [tex]\( t = 8 \)[/tex] seconds, the height [tex]\( h(8) = 6.4 \)[/tex].
- At [tex]\( t = 10 \)[/tex] seconds, the height [tex]\( h(10) = -90 \)[/tex].
- Since neither 6.4 nor -90 is equal to the initial height (0 meters), the ball is not at the same height as the building between 8 and 10 seconds.
Result: False
2. The height of the ball decreases and then increases.
- The ball's height increases initially from [tex]\( t = 0 \)[/tex] to [tex]\( t = 4 \)[/tex] (from 0 to 81.6).
- Afterwards, the height decreases from [tex]\( t = 4 \)[/tex] to [tex]\( t = 8 \)[/tex] (from 81.6 to 6.4).
- There is no point after [tex]\( t = 8 \)[/tex] where the height again increases; instead, it further decreases.
- Therefore, the ball's height does not exhibit a pattern of decreasing and then increasing.
Result: True
3. The ball reaches its maximum height about 4 seconds after it is thrown.
- At [tex]\( t = 4 \)[/tex] seconds, the height [tex]\( h(4) = 81.6 \)[/tex], which is the highest value in the data provided.
- Thus, the ball does reach its maximum height at about 4 seconds.
Result: True
4. The ball hits the ground between 8 and 10 seconds after it is thrown.
- At [tex]\( t = 8 \)[/tex] seconds, the height [tex]\( h(8) = 6.4 \)[/tex] (the ball is still above the ground).
- At [tex]\( t = 10 \)[/tex] seconds, the height [tex]\( h(10) = -90 \)[/tex], which indicates the ball has gone below the ground level (since height is negative).
- Thus, the ball does hit the ground between 8 and 10 seconds.
Result: True
5. The height of the building is 81.6 meters.
- The height at [tex]\( t = 0 \)[/tex] seconds, when the ball is just thrown, is 0 meters.
- Hence, the actual height of the building is 0 meters.
Result: False
### Summary:
- False: The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- True: The height of the ball decreases and then increases.
- True: The ball reaches its maximum height about 4 seconds after it is thrown.
- True: The ball hits the ground between 8 and 10 seconds after it is thrown.
- False: The height of the building is 81.6 meters.
Therefore, the true statements are:
- The height of the ball decreases and then increases.
- The ball reaches its maximum height about 4 seconds after it is thrown.
- The ball hits the ground between 8 and 10 seconds after it is thrown.
### Data Analysis:
The height values are given as follows:
- at [tex]\( t = 0\, \text{seconds}: h(t) = 0\)[/tex]
- at [tex]\( t = 2\, \text{seconds}: h(t) = 60.4\)[/tex]
- at [tex]\( t = 4\, \text{seconds}: h(t) = 81.6\)[/tex]
- at [tex]\( t = 6\, \text{seconds}: h(t) = 63.6\)[/tex]
- at [tex]\( t = 8\, \text{seconds}: h(t) = 6.4\)[/tex]
- at [tex]\( t = 10\, \text{seconds}: h(t) = -90\)[/tex]
- at [tex]\( t = 12\, \text{seconds}: h(t) = -225.6\)[/tex]
### Evaluation of Statements:
1. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- At [tex]\( t = 8 \)[/tex] seconds, the height [tex]\( h(8) = 6.4 \)[/tex].
- At [tex]\( t = 10 \)[/tex] seconds, the height [tex]\( h(10) = -90 \)[/tex].
- Since neither 6.4 nor -90 is equal to the initial height (0 meters), the ball is not at the same height as the building between 8 and 10 seconds.
Result: False
2. The height of the ball decreases and then increases.
- The ball's height increases initially from [tex]\( t = 0 \)[/tex] to [tex]\( t = 4 \)[/tex] (from 0 to 81.6).
- Afterwards, the height decreases from [tex]\( t = 4 \)[/tex] to [tex]\( t = 8 \)[/tex] (from 81.6 to 6.4).
- There is no point after [tex]\( t = 8 \)[/tex] where the height again increases; instead, it further decreases.
- Therefore, the ball's height does not exhibit a pattern of decreasing and then increasing.
Result: True
3. The ball reaches its maximum height about 4 seconds after it is thrown.
- At [tex]\( t = 4 \)[/tex] seconds, the height [tex]\( h(4) = 81.6 \)[/tex], which is the highest value in the data provided.
- Thus, the ball does reach its maximum height at about 4 seconds.
Result: True
4. The ball hits the ground between 8 and 10 seconds after it is thrown.
- At [tex]\( t = 8 \)[/tex] seconds, the height [tex]\( h(8) = 6.4 \)[/tex] (the ball is still above the ground).
- At [tex]\( t = 10 \)[/tex] seconds, the height [tex]\( h(10) = -90 \)[/tex], which indicates the ball has gone below the ground level (since height is negative).
- Thus, the ball does hit the ground between 8 and 10 seconds.
Result: True
5. The height of the building is 81.6 meters.
- The height at [tex]\( t = 0 \)[/tex] seconds, when the ball is just thrown, is 0 meters.
- Hence, the actual height of the building is 0 meters.
Result: False
### Summary:
- False: The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
- True: The height of the ball decreases and then increases.
- True: The ball reaches its maximum height about 4 seconds after it is thrown.
- True: The ball hits the ground between 8 and 10 seconds after it is thrown.
- False: The height of the building is 81.6 meters.
Therefore, the true statements are:
- The height of the ball decreases and then increases.
- The ball reaches its maximum height about 4 seconds after it is thrown.
- The ball hits the ground between 8 and 10 seconds after it is thrown.
