Answer: To determine the total distance traveled by the volleyball ball before coming to rest, we can sum up the distances of each rebound. The ball rebounds 1/4 of the distance of the previous ball for each rebound. Let's calculate the distances traveled for each rebound until the ball comes to rest.
First rebound:
The ball is dropped from a height of 4 meters, so it reaches the ground and rebounds back up to a height of 4 * (1/4) = 1 meter.
Distance traveled in the first rebound:
4 meters (downward) + 1 meter (upward) = 5 meters
Second rebound:
The ball was at a height of 1 meter, and it rebounds 1/4 of this distance, which is 1 * (1/4) = 0.25 meters.
Distance traveled in the second rebound:
1 meter (downward) + 0.25 meters (upward) = 1.25 meters
Third rebound:
The ball was at a height of 0.25 meters, and it rebounds 1/4 of this distance, which is 0.25 * (1/4) = 0.0625 meters.
Distance traveled in the third rebound:
0.25 meters (downward) + 0.0625 meters (upward) = 0.3125 meters
The ball continues to rebound with decreasing distances, approaching zero. To find the total distance traveled before coming to rest, we can sum up the distances from each rebound.
Total distance traveled:
5 meters + 1.25 meters + 0.3125 meters + ...
This is an infinite geometric series with a common ratio of 1/4. The sum of an infinite geometric series can be calculated using the formula:
Sum = a / (1 - r)
where a is the first term and r is the common ratio.
Plugging in the values:
a = 5 meters (distance of the first rebound)
r = 1/4
Sum = 5 / (1 - 1/4)
Sum = 5 / (3/4)
Sum = 5 * (4/3)
Sum = 20/3 ≈ 6.67 meters
Therefore, the volleyball ball travels approximately 6.67 meters before coming to rest.