To determine the correct statement comparing the line of best fit with the actual data, let's follow these steps:
1. Equation of the Line of Best Fit:
Given the line of best fit equation:
[tex]\[ h = -21.962x + 114.655 \][/tex]
2. Time when Height is 40 Meters:
We want to evaluate the height according to the line of best fit at [tex]\( x = 3.5 \)[/tex] seconds.
3. Calculate the Predicted Height:
Substitute [tex]\( x = 3.5 \)[/tex] into the equation:
[tex]\[ h = -21.962(3.5) + 114.655 \][/tex]
4. Predicted Height Calculation:
Simplifying the equation:
[tex]\[ h = -76.867 + 114.655 \][/tex]
[tex]\[ h = 37.788 \][/tex]
So, the height predicted by the line of best fit at [tex]\( t = 3.5 \)[/tex] seconds is approximately 37.788 meters.
5. Compare with Actual Data:
From the table, the actual height at [tex]\( t = 3.5 \)[/tex] seconds is 40.0 meters.
The predicted height (37.788 meters) is not the same as the actual height (40.0 meters), and the difference is more than 0.1 meters.
6. Evaluate the Statements:
- The first statement ("the object would have hit the ground 0.6 seconds later than the actual time") needs to be checked by solving [tex]\( -21.962x + 114.655 = 0 \)[/tex]. This isn't directly addressed here.
- The second statement ("the object was dropped from a lower height") implies the initial height is lower in the best-fit line than in the data, which needs the initial height without evidence here.
- The third statement ("reaches a height of 40 meters after 3.5 seconds") is false, since the predicted height is 37.788 meters, not 40 meters.
- The fourth statement ("the line of best fit predicts a height of 4 meters greater") is not general and unlikely correct.
From this, we see that the correct statement from these evaluations is:
- The line of best fit does not correctly predict that the object reaches a height of 40 meters after 3.5 seconds.
