This is a classic geometric series problem involving a bouncing ball. To find the total distance the ball travels in 15 bounces, we need to consider both the distances it falls and the distances it rebounds.

Step-by-step solution:

1. Initial drop: The ball is dropped from a height of 16 feet, so the first fall is 16 feet.

2. Rebound heights: After each drop, the ball rebounds to 80% of the height it fell from. Therefore, the height of the first rebound is $16 \times 0.8 = 12$ feet.

3. Distance traveled for each bounce: For each bounce, the ball falls down and then rebounds up. So for each bounce, the total distance covered is:

   • Fall distance = the current height it falls from.
   • Rebound distance = 80% of the fall height.

   This process continues until we reach the 15th bounce.

Total distance:

• The ball travels 16 feet on the first fall.
• On each subsequent bounce, the ball travels the sum of the falling and rebounding distances.

The distances for the subsequent bounces follow a geometric series:

• First fall = 16 feet.
• For the bounces:
  • Fall + rebound: $2 \times 16 \times (0.8^n)$, where $n$ is the bounce number.

The total distance $D$ can be found as follows:

$D = 16 + 2 \times \sum_{n=1}^{14} \left(16 \times 0.8^n \right)$

We can now compute this geometric series.

Let me calculate that for you.The total distance traveled by the ball in 15 bounces is approximately 138.37 feet.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How would the total distance change if the rebound ratio was 90% instead of 80%?
2. What is the distance traveled after only 10 bounces?
3. How would you calculate the total distance if the ball continued bouncing indefinitely?
4. What would be the total distance if the initial height was 20 feet instead of 16 feet?
5. How does the rebound percentage affect the convergence of the total distance?

Tip: Geometric series are helpful when analyzing repeated actions with a constant ratio, like bouncing or financial investments!