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The problem you've provided describes a ball being tossed to a height of 4 meters and rebounding to 40% (or 0.40) of its previous height on each bounce. You are asked to find the total distance the ball has traveled after the fifth bounce.

### Breakdown of the problem:
1. **Initial height** = 4 meters.
2. **Rebound factor** = 0.40 (after every bounce, it reaches 40% of the previous height).
3. You need to calculate the total distance the ball has traveled after it has rebounded up to the fifth bounce.

### Steps:
The ball goes through two phases:
- **Upward motion** (when it rebounds to a fraction of its previous height).
- **Downward motion** (when it falls back to the ground).

The total distance can be found by adding all the distances the ball has traveled.

1. **Initial fall**: The ball falls from a height of 4 meters (downward motion).
2. **First rebound**: The ball rebounds to 40% of the height, i.e., \( 4 \times 0.40 = 1.6 \) meters (upward).
3. **First fall**: The ball then falls from that height of 1.6 meters (downward).
4. **Second rebound**: The ball rebounds to 40% of 1.6 meters, which is \( 1.6 \times 0.40 = 0.64 \) meters (upward).
5. **Second fall**: The ball falls from that height (0.64 meters).
6. This process continues until the fifth rebound.

The total distance will be the sum of all upward and downward motions.

Let me calculate this for you.The total distance the ball has traveled until the fifth rebound is approximately **9.20 meters**.

This includes the initial fall, as well as all the upward and downward motions up to the fifth rebound.

Would you like more details, or do you have any other questions?

Here are 5 related questions:
1. How would the total distance change if the ball only rebounded to 30% of its previous height?
2. What is the general formula for calculating total distance after \(n\) bounces?
3. What if the ball was initially tossed to a height of 6 meters? How would that affect the total distance?
4. Can you calculate the total distance if the ball never stopped bouncing (i.e., infinite bounces)?
5. How does this problem relate to the concept of geometric series in mathematics?

**Tip:** When solving such problems, identifying the geometric sequence in the rebounding heights can help you apply formulas for sums of infinite or finite series efficiently.

A ball tossed to a height of 4 meters rebounds to 40% (0.40 in decimal) of its previous height. What is the total height/distance the ball has rebounded until the fifth time?

Discover how to calculate the total distance a ball travels after being tossed to a height of 4 meters and rebounding to 40% of its previous height up to the fifth rebound. The solution involves understanding geometric progressions and summing multiple series terms.

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