To find the distance traveled by the ball up to the time it strikes the ground for the 8th time, we need to calculate the total distance covered during all the bounces.
Given: Initial height (h) = 1458 meters Bounce ratio (b) = 2/3
The ball bounces to a fraction of the height from which it last fell, so we can calculate the height of each bounce as follows:
First bounce: h Second bounce: b * h Third bounce: b * b * h Fourth bounce: b * b * b * h Fifth bounce: b * b * b * b * h Sixth bounce: b * b * b * b * b * h Seventh bounce: b * b * b * b * b * b * h Eighth bounce: b * b * b * b * b * b * b * h
To find the total distance traveled, we sum up all these heights:
Total distance traveled = h + b * h + b * b * h + b * b * b * h + b * b * b * b * h + b * b * b * b * b * h + b * b * b * b * b * b * h + b * b * b * b * b * b * b * h
Now, let's calculate the total distance covered:
Total distance traveled = h * (1 + b + b^2 + b^3 + b^4 + b^5 + b^6 + b^7)
where b = 2/3.
Substituting the values:
Total distance traveled = 1458 * (1 + (2/3) + (2/3)^2 + (2/3)^3 + (2/3)^4 + (2/3)^5 + (2/3)^6 + (2/3)^7)
Evaluating this expression:
Total distance traveled ≈ 1458 * (1 + 2/3 + 4/9 + 8/27 + 16/81 + 32/243 + 64/729 + 128/2187)
Total distance traveled ≈ 1458 * (1 + 2/3 + 4/9 + 8/27 + 16/81 + 32/243 + 64/729 + 128/2187)
Total distance traveled ≈ 1458 * (1 + 0.6667 + 0.4444 + 0.2963 + 0.1975 + 0.1317 + 0.0879 + 0.0586)
Total distance traveled ≈ 1458 * 3.8821
Total distance traveled ≈ 5660.7998 meters (rounded to four decimal places)
Therefore, up to the time it strikes the ground for the 8th time, the rubber ball has traveled approximately 5660.7998 meters.