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."
