To find the approximate distance the ball travels before coming to rest, we need to sum up the distances covered during each bounce.
Given that each bounce is about 4/5 as high as the preceding one, we can say that the height of each bounce is 4/5 of the previous bounce.
Let's calculate the height of each bounce using the formula:
Height of bounce = (4/5) * previous bounce height
Given that the initial drop height is 20 ft, we can write the heights of each bounce as follows:
1st bounce: 20 ft
2nd bounce: (4/5) * 20 ft
3rd bounce: (4/5) * (4/5) * 20 ft
4th bounce: (4/5) * (4/5) * (4/5) * 20 ft
The general formula to calculate the height of the nth bounce is:
Height of nth bounce = (4/5)^(n-1) * 20 ft
Now, we need to add up the distances traveled during each bounce. The distance traveled is twice the height of each bounce since the ball covers the same distance going up and coming back down.
So, the distance traveled during each bounce is:
Distance of bounce = 2 * height of bounce
Using the height formula mentioned above, the distance of each bounce can be written as:
Distance of nth bounce = 2 * (4/5)^(n-1) * 20 ft
Now, let's calculate the total distance covered by summing up all the distances of each bounce until it comes to rest. We need to find the last bounce where the height becomes negligible.
Let's calculate the height of the bounce until it becomes negligible when it is less than 1 ft.
(4/5)^(n-1) * 20 ft < 1 ft
Dividing both sides by 20 ft:
(4/5)^(n-1) < 1/20
Taking the base 4/5 logarithm of both sides:
(n-1) < log(1/20) / log(4/5)
Simplifying:
(n-1) < log(1/20) / log(4/5)
(n-1) < -2.301 / -0.322
(n-1) < 7.13
Since n has to be a positive integer, the last bounce will be when n = 8.
Now, let's calculate the total distance traveled:
Total distance = Distance of 1st bounce + Distance of 2nd bounce + ... + Distance of 8th bounce
Total distance = [2 * (4/5)^(0) * 20] + [2 * (4/5)^(1) * 20] + ... + [2 * (4/5)^(7) * 20]
Total distance = 40 + 32 + 25.6 + 20.48 + 16.38 + 13.10 + 10.48 + 8.38
Total distance ≈ 166.42 ft
Therefore, the approximate distance the ball travels before coming to rest is approximately 166.42 ft.