Let's analyze the problem and break it down step by step.

Problem Summary:

A rubber ball is dropped from a height of 80 feet, and each time it rebounds, it bounces up to exactly half the previous height. The task involves determining how far the ball will have traveled after the 7th bounce, if it is caught at the top of that bounce.

Steps to solve the problem:

A. Diagram

• The problem asks for a diagram showing how the ball bounces up and down. This can be represented visually by drawing arrows for the ball's initial drop (80 feet), then subsequent rebounds (halving each time). Each "upward" and "downward" movement should be shown, with diminishing heights.

B. Type of Sequence

• The distances traveled by the ball form a geometric sequence, because each successive distance is half of the previous one.
• The sequence involves both the upward and downward movements. For the first bounce, the ball falls 80 feet, then bounces up 40 feet, then falls 40 feet again, and so on.

Sequence:

• The sequence of distances (just the bounces or combined movements) would be: $80, 40, 40, 20, 20, 10, 10, 5, 5, \dots$

C. Identify Variables:

• $a_1 = 80$ feet (initial drop).
• $n = 7$ (we want to stop at the 7th bounce).
• $r = \frac{1}{2}$ (common ratio, because the height halves each time).
• $S_n$ is the total distance traveled up to and including the 7th bounce, which we need to compute.

D. Formula to Use:

The total distance traveled after 7 bounces includes both the fall and rebound for each bounce, except the last one where we only count the upward movement.

For a geometric series, the sum formula is: $S_n = a_1 \frac{1 - r^n}{1 - r}$ But since the ball goes both up and down after each bounce (except the last bounce where it’s only up), we will need to calculate the sum of the geometric sequence for both up and down movements.

Let me compute the total distance traveled.The total distance the ball travels after it reaches the top of the seventh bounce is 237.5 feet. This includes both the ball's falling and rebounding movements.

Summary:

• A. Diagram (visual representation of the ball bouncing up and down).
• B. The sequence formed is a geometric sequence.
• C. Variables identified:
  • $a_1 = 80$ feet,
  • $n = 7$,
  • $r = \frac{1}{2}$,
  • $S_n = 237.5$ feet.
• D. The formula used is the sum of a geometric series, adjusted to account for the upward and downward movements.

Do you have any questions or would you like further clarification on any part of the solution?

Here are 5 related questions:

1. How would the distance change if the ball bounced back to one-third of the previous height instead?
2. What would happen to the total distance if the ball was caught after the 10th bounce instead of the 7th?
3. How would the geometric sequence change if the initial drop was from 100 feet?
4. What is the sum of the geometric series after infinite bounces?
5. Can you derive a general formula for any number of bounces $n$?

Tip: When solving geometric series problems, identifying the common ratio $r$ early on helps greatly in setting up the correct formula.