Final answer:
The ball travels a total distance of 1200 meters before coming to rest, calculated using the sum of an infinite geometric series with the first term being 300 meters and the common ratio being 4/5.
Step-by-step explanation:
To determine the total distance the ball travels before coming to rest, we need to consider both the descent and ascent distances for each bounce. Since the ball rebounds to 4/5ths of the previous height, we have a geometric series where the first term is 300 m (the initial drop) and the common ratio is 4/5. The total distance traveled by the ball is the sum of the infinite geometric series:
- First drop = 300 m
- First rebound = 300 m * (4/5) = 240 m
- Second drop = 240 m
- Second rebound = 240 m * (4/5) = 192 m
- and so on...
We calculate the sum of the infinite series using the formula:
S = a / (1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio.
In this case, a = 300 m and r = 4/5. Plugging these values into the formula:
S = 300 / (1 - 4/5) = 300 / (1/5) = 300 * 5 = 1500 m
However, this includes both the ascent and descent of each bounce. Since the initial drop doesn't have a preceding ascent, we must subtract one initial height of 300 m to obtain the total distance traveled:
Total distance = 1500 m - 300 m = 1200 m
Therefore, the ball travels a total distance of 1200 meters.