Question
A bouncing ball reaches a height of 27 feet at its first peak, 18 feet at its second peak, and 12 feet at its third peak. Describe how a sequence can be used to determine the height of the ball when it reaches its fourth peak.
Answers
GPT-4o mini
To determine the height of the ball when it reaches its fourth peak, we can analyze the sequence of the heights that the ball reaches after each bounce. The heights at the peaks are as follows:
1. First peak: 27 feet
2. Second peak: 18 feet
3. Third peak: 12 feet
To find a pattern, we can observe the changes in height between each peak:
- From the first peak to the second peak: 27 feet to 18 feet
- Change: 27 - 18 = 9 feet
- From the second peak to the third peak: 18 feet to 12 feet
- Change: 18 - 12 = 6 feet
Next, we can examine the pattern of the changes:
- The first change (from 27 to 18) is a decrease of 9 feet.
- The second change (from 18 to 12) is a decrease of 6 feet.
Now let's determine if there is a consistent pattern in these changes. The decrease from the first to the second peak is by 9 feet, and from the second to the third peak is by 6 feet. The decrease itself seems to be decreasing by 3 feet each time (9 - 6 = 3).
If we assume the pattern continues, we can expect the change from the third peak to the fourth peak to also decrease by 3 feet from the previous change (6 feet). Thus, the expected decrease from the third peak to the fourth peak would be:
6 - 3 = 3 feet decrease.
Now we can apply this change to the height of the third peak (12 feet):
Height at the fourth peak = Height at the third peak - Decrease from third to fourth peak
Height at the fourth peak = 12 - 3 = 9 feet.
Therefore, the height of the ball when it reaches its fourth peak is expected to be **9 feet**.
In summary, the sequence of heights is decreasing in a pattern, and using the observed changes, we can predict that the height at the fourth peak will be 9 feet.
1. First peak: 27 feet
2. Second peak: 18 feet
3. Third peak: 12 feet
To find a pattern, we can observe the changes in height between each peak:
- From the first peak to the second peak: 27 feet to 18 feet
- Change: 27 - 18 = 9 feet
- From the second peak to the third peak: 18 feet to 12 feet
- Change: 18 - 12 = 6 feet
Next, we can examine the pattern of the changes:
- The first change (from 27 to 18) is a decrease of 9 feet.
- The second change (from 18 to 12) is a decrease of 6 feet.
Now let's determine if there is a consistent pattern in these changes. The decrease from the first to the second peak is by 9 feet, and from the second to the third peak is by 6 feet. The decrease itself seems to be decreasing by 3 feet each time (9 - 6 = 3).
If we assume the pattern continues, we can expect the change from the third peak to the fourth peak to also decrease by 3 feet from the previous change (6 feet). Thus, the expected decrease from the third peak to the fourth peak would be:
6 - 3 = 3 feet decrease.
Now we can apply this change to the height of the third peak (12 feet):
Height at the fourth peak = Height at the third peak - Decrease from third to fourth peak
Height at the fourth peak = 12 - 3 = 9 feet.
Therefore, the height of the ball when it reaches its fourth peak is expected to be **9 feet**.
In summary, the sequence of heights is decreasing in a pattern, and using the observed changes, we can predict that the height at the fourth peak will be 9 feet.