Final answer:
The total distance traveled by a ball dropped from 20 meters that rebounds to four-fifths of its previous height involves an infinite geometric series. The first term and the common ratio 'r' are used in the formula S = a / (1 - r) to calculate the sum of the series. This sum is then doubled (to account for both descent and ascent after the first drop) and added to the initial 20 meters to give the overall distance.
Step-by-step explanation:
To calculate the total distance a ball travels before coming to rest, assuming it rebounds to a height of four-fifths of its previous height, we consider the first drop from 20 meters and the subsequent bounces. The following is a geometric series, where each term after the initial drop is 4/5 times the previous term:
- First drop = 20 meters
- First rebound = 16 meters (0.8 * 20)
- Second rebound = 12.8 meters (0.8 * 16)
- ... and so on.
Each bounce has both a descent and an ascent (except the first drop), so the distance of each bounce is counted twice. To find the total distance traveled, the summation can be expressed as:
Total distance
= 20m + 2 * (16m + 0.8 * 16m + 0.8^2 * 16m + ...)
Since this is a convergent geometric series, we use the formula for the sum of an infinite geometric series, S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. Therefore, the total distance is:
Total distance
= 20m + 2 * (16m / (1 - 0.8))
After calculating, we add the initial 20 meters to the result to account for the first drop, giving us the overall total distance the ball travels.