Question

the table shows the height, in meters, of an object that is dropped as time passes until the object hits the ground.
falling object

time (sec), x	0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5
height (m), h	100	98.8	95.1	89.0	80.4	69.4	55.0	40.0	21.0	0

a line of best fit for the data is represented by ( h = -21.962x + 114.655 ). which statement compares the line of best fit with the actual data given by the table?

• 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.
• according to the line of best fit, the object was dropped from a lower height.
• the line of best fit correctly predicts that the object reaches a height of 40 meters after 3.5 seconds.
• the line of best fit predicts a height of 4 meters greater than the actual height for any time given in the table.

Explanation:

Step1: Analyze when the object hits the ground

To find when the object hits the ground according to the line of best fit, set \( h = 0 \) in \( h=-21.962x + 114.655 \).

$$ 0=-21.962x + 114.655 $$

$$ 21.962x = 114.655 $$

$$ x=\frac{114.655}{21.962}\approx5.22 $$

From the table, the object hits the ground at \( x = 4.6 \) seconds. The difference is \( 5.22 - 4.6 = 0.62\approx0.6 \) seconds. So the first option seems possible.

Step2: Check the initial height (when \( x = 0 \))

For the line of best fit, when \( x = 0 \), \( h = 114.655 \) meters. From the table, initial height is 100 meters. So the second option is wrong (line of best fit says higher initial height).

Step3: Check height at \( x = 3.5 \)

For the line of best fit, \( h=-21.962(3.5)+114.655=-76.867 + 114.655 = 37.788\approx37.8 \) meters. The table says 40 meters. So the third option is wrong.

Step4: Check if prediction is always 4 meters greater

At \( x = 0.5 \), line of best fit: \( h=-21.962(0.5)+114.655=-10.981 + 114.655 = 103.674 \). Actual height is 98.8. Difference is \( 103.674 - 98.8 = 4.874\approx4.9 \), not 4. So fourth option is wrong.

Answer:

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.