To find the total distance traveled by the ball before it comes to rest, we need to consider both the distances it falls and the distances it bounces back up.

1. Initial fall: The ball falls from 10 meters.
2. Rebounds: Each time it rebounds, it reaches 80% of the previous height.

Let's break it down step-by-step:

1. First fall: 10 meters.
2. First bounce up: $10 \times 0.8 = 8$ meters.
3. Second fall: 8 meters.
4. Second bounce up: $8 \times 0.8 = 6.4$ meters.
5. Third fall: 6.4 meters.
6. Third bounce up: $6.4 \times 0.8 = 5.12$ meters.
7. And so on...

The total distance traveled by the ball is the sum of the distances it falls and bounces back up.

We observe that after the initial fall, each bounce up and fall down form a pair of equal distances. Therefore, we can use a geometric series to find the total distance.

Let the initial height be $h = 10$ meters, and the rebound ratio be $r = 0.8$.

Total distance traveled:

$\text{Total distance} = 10 + 2 \sum_{n=1}^{\infty} 10 \times (0.8)^n$

The series $\sum_{n=1}^{\infty} 10 \times (0.8)^n$ is a geometric series with the first term $a = 10 \times 0.8 = 8$ and the common ratio $r = 0.8$.

The sum of an infinite geometric series is given by:

$S = \frac{a}{1 - r}$

Here,

$a = 8$ $r = 0.8$

So,

$S = \frac{8}{1 - 0.8} = \frac{8}{0.2} = 40$

Therefore, the total distance is:

$\text{Total distance} = 10 + 2 \times 40 = 10 + 80 = 90 \text{ meters}$

Thus, the total distance traveled by the ball before it comes to rest is $90$ meters.

Would you like more details on any part of the solution or have any questions?

Here are 8 related questions for further exploration:

1. What would be the total distance if the initial height was 20 meters?
2. How would the total distance change if the rebound ratio was 70% instead of 80%?
3. What is the general formula for the total distance traveled given any initial height and rebound ratio?
4. How does the number of bounces affect the total distance traveled?
5. If the ball only rebounded 50% of its height each time, what would be the total distance?
6. Can you derive the formula for the distance of the $n$-th bounce?
7. What if the ball rebounded 100% of its height every time, what would be the total distance then?
8. How would air resistance affect the total distance traveled by the ball?

Tip: For solving problems involving infinite geometric series, remember the sum formula $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.