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Sagot :
First, let's analyze the given data and the line of best fit:
The actual data from the table shows that the heights of the object are recorded at specific times, from 0 seconds to 4.6 seconds.
The given equation of the line of best fit is:
[tex]\[ h = -21.962x + 114.655 \][/tex]
We'll evaluate the statements one by one using the given results:
1. Statement 1: According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground.
- Actual time when the object hits the ground is at 4.6 seconds.
- Predicted time using the line of best fit when [tex]\(h = 0\)[/tex] is:
[tex]\[ 0 = -21.962x + 114.655 \implies x = \frac{114.655}{21.962} \approx 5.22 \, \text{seconds} \][/tex]
The difference between the predicted time and the actual time is:
[tex]\[ 5.22 \, \text{seconds} - 4.6 \, \text{seconds} = 0.62 \, \text{seconds} \][/tex]
So, this statement is true: the object would have hit the ground approximately 0.6 seconds later than the actual time.
2. Statement 2: According to the line of best fit, the object was dropped from a lower height.
- The intercept of the line of best fit is 114.655, indicating the height from which the object was dropped according to the model.
- The actual height from which the object was dropped is 100 meters.
Since [tex]\(114.655 > 100\)[/tex], the predicted height is actually higher than the actual height. Thus, this statement is false.
3. Statement 3: The line of best fit correctly predicts that the object reaches a height of 40 meters after 3.5 seconds.
- According to the line of best fit, the height at [tex]\(x = 3.5\)[/tex] seconds is:
[tex]\[ h = -21.962 \times 3.5 + 114.655 \approx 37.788 \, \text{meters} \][/tex]
- The actual height from the table at [tex]\(x = 3.5\)[/tex] seconds is 40 meters.
So, this statement is false because 37.788 meters is not equal to 40 meters.
4. Statement 4: The line of best fit predicts a height of 4 meters greater than the actual height for any time given in the table.
- For this statement, we need to look at the height differences given by the results:
[tex]\[ \text{Height differences} = [-14.655, -4.874, 2.407, 7.288, 9.669, 9.650, 7.131, 2.212, -5.207, -13.63] \][/tex]
- The differences vary and are sometimes positive, sometimes negative, and not consistently equal to 4 meters.
Therefore, this statement is false.
### Conclusion:
The correct statement according to the comparisons made is:
"According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground."
The actual data from the table shows that the heights of the object are recorded at specific times, from 0 seconds to 4.6 seconds.
The given equation of the line of best fit is:
[tex]\[ h = -21.962x + 114.655 \][/tex]
We'll evaluate the statements one by one using the given results:
1. Statement 1: According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground.
- Actual time when the object hits the ground is at 4.6 seconds.
- Predicted time using the line of best fit when [tex]\(h = 0\)[/tex] is:
[tex]\[ 0 = -21.962x + 114.655 \implies x = \frac{114.655}{21.962} \approx 5.22 \, \text{seconds} \][/tex]
The difference between the predicted time and the actual time is:
[tex]\[ 5.22 \, \text{seconds} - 4.6 \, \text{seconds} = 0.62 \, \text{seconds} \][/tex]
So, this statement is true: the object would have hit the ground approximately 0.6 seconds later than the actual time.
2. Statement 2: According to the line of best fit, the object was dropped from a lower height.
- The intercept of the line of best fit is 114.655, indicating the height from which the object was dropped according to the model.
- The actual height from which the object was dropped is 100 meters.
Since [tex]\(114.655 > 100\)[/tex], the predicted height is actually higher than the actual height. Thus, this statement is false.
3. Statement 3: The line of best fit correctly predicts that the object reaches a height of 40 meters after 3.5 seconds.
- According to the line of best fit, the height at [tex]\(x = 3.5\)[/tex] seconds is:
[tex]\[ h = -21.962 \times 3.5 + 114.655 \approx 37.788 \, \text{meters} \][/tex]
- The actual height from the table at [tex]\(x = 3.5\)[/tex] seconds is 40 meters.
So, this statement is false because 37.788 meters is not equal to 40 meters.
4. Statement 4: The line of best fit predicts a height of 4 meters greater than the actual height for any time given in the table.
- For this statement, we need to look at the height differences given by the results:
[tex]\[ \text{Height differences} = [-14.655, -4.874, 2.407, 7.288, 9.669, 9.650, 7.131, 2.212, -5.207, -13.63] \][/tex]
- The differences vary and are sometimes positive, sometimes negative, and not consistently equal to 4 meters.
Therefore, this statement is false.
### Conclusion:
The correct statement according to the comparisons made is:
"According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground."
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