A ball is dropped from the top of a 25-m ladder. in each bounce, the ball reaches a vertical height that 3/5 the previous vertical height. determine the total vertical distance travelled by the ball when it contacts the ground for the sixth time. express your answer to the nearest tenth of a meter.
6 round trips, less the 1st 25m upwards
a = 25
r = 3/5
S6 = a(1-r^6)/(1-5)
so, we get 25(1 - (3/5)^6)/(1 - 3/5) - 25 = _____
rats. round trips.
2*25(1 - (3/5)^6)/(1 - 3/5) - 25
To determine the total vertical distance traveled by the ball when it contacts the ground for the sixth time, we need to calculate the sum of the distances traveled during each bounce.
1. The initial height of the ball is 25 meters.
2. In each bounce, the ball reaches a height that is 3/5 (or 0.6) times the previous height.
Using this information, we can set up a geometric sequence to represent the heights reached during each bounce.
First, we need to determine the common ratio of the geometric sequence:
Common Ratio (r) = 0.6
Now, let's calculate the heights reached during each bounce:
- First bounce: Initial height = 25 meters
- Second bounce: Height = 25 * 0.6 = 15 meters
- Third bounce: Height = 15 * 0.6 = 9 meters
- Fourth bounce: Height = 9 * 0.6 = 5.4 meters
- Fifth bounce: Height = 5.4 * 0.6 = 3.24 meters
- Sixth bounce: Height = 3.24 * 0.6 = 1.944 meters
Now, let's calculate the total vertical distance traveled by the ball:
Total Distance = Initial height + Height of first bounce + Height of second bounce + ... + Height of sixth bounce
Total Distance = 25 + 15 + 9 + 5.4 + 3.24 + 1.944 = 59.584 meters
Therefore, the total vertical distance traveled by the ball when it contacts the ground for the sixth time is approximately 59.6 meters (rounded to the nearest tenth).
To find the total vertical distance traveled by the ball when it contacts the ground for the sixth time, we need to sum up the distances traveled in each bounce.
Given that the ball reaches a vertical height that is 3/5 of the previous vertical height in each bounce, we can use a geometric sequence to calculate the distances. The formula for the sum of a geometric sequence is:
S = a(1 - r^n) / (1 - r)
where:
S = sum of the geometric sequence
a = first term of the sequence
r = common ratio of the sequence
n = number of terms in the sequence
In this case, the first term (a) is 25 meters (height of the ladder). The common ratio (r) is 3/5 since the ball reaches 3/5 of the previous height in each bounce. The number of terms (n) is 6 since we want to calculate the total distance for the first six bounces.
Using the formula, we can calculate the sum:
S = 25(1 - (3/5)^6) / (1 - 3/5)
Simplifying the equation:
S = 25(1 - 0.04665625) / (0.2)
S = 25(0.95334375) / (0.2)
S = 119.16875 / 0.2
S = 595.84375 meters
Therefore, the total vertical distance traveled by the ball when it contacts the ground for the sixth time is approximately 595.8 meters, rounded to the nearest tenth.