## To calculate the total distance traveled by the ball when it hits the ground for the fifth time, we need to determine the distance traveled during each bounce and add them up. Let's break down the problem to find a pattern.

The ball is dropped from a height of 3m, so the first bounce reaches 60% of 3m, which is 1.8m. The ball then rebounds back up to a certain height and falls again.

For each subsequent bounce, the ball reaches 60% of its previous height. So, the second bounce will reach 60% of 1.8m, which is 1.08m. Similarly, the third bounce will reach 60% of 1.08m, which is 0.648m, and so on.

We need to find the total distance traveled when the ball hits the ground for the fifth time. To do that, we add up the heights reached during each bounce.

During the first bounce, the ball travels a distance of 3m (from the initial drop) plus the distance traveled during the ascent and descent of the ball. Since the ball reaches a height of 1.8m during the first bounce, the total distance traveled during the first bounce is twice that distance, or 2 * 1.8m = 3.6m.

Now, let's calculate the total distance traveled during the subsequent bounces:

- Second bounce: The height reached is 60% of 1.8m, which is 1.08m. The total distance traveled during the second bounce is twice that distance, or 2 * 1.08m = 2.16m.

- Third bounce: The height reached is 60% of 1.08m, which is 0.648m. The total distance traveled during the third bounce is twice that distance, or 2 * 0.648m = 1.296m.

- Fourth bounce: The height reached is 60% of 0.648m, which is 0.3888m. The total distance traveled during the fourth bounce is twice that distance, or 2 * 0.3888m = 0.7776m.

- Fifth bounce: The height reached is 60% of 0.3888m, which is 0.2333m (rounded to 4 decimal places). The total distance traveled during the fifth bounce is twice that distance, or 2 * 0.2333m = 0.4666m (rounded to 4 decimal places).

To find the total distance traveled by the ball when it hits the ground for the fifth time, we add up all the distances traveled during each bounce:

3.6m + 2.16m + 1.296m + 0.7776m + 0.4666m = 8.2992m

Therefore, the total distance traveled by the ball when it hits the ground for the fifth time is approximately 8.2992 meters.