The problem describes a rubber ball dropped from a height of 80 feet that bounces to half the previous height on each bounce. You are asked to find the total distance traveled by the ball after it reaches the top of the seventh bounce.

Step-by-Step Breakdown:

1. Initial Drop:

The ball falls 80 feet to the ground, so the first part of the journey is straightforward:

• Distance traveled during the initial drop = 80 feet.

2. Bounces:

The ball then bounces up and down, and the height of each bounce follows a geometric sequence. Each time, the ball bounces to half the height of the previous bounce.

The heights of the bounces form a geometric sequence where the first term is 80 feet, and the common ratio $r = \frac{1}{2}$.

• First bounce height = 40 feet (half of 80 feet)
• Second bounce height = 20 feet (half of 40 feet)
• Third bounce height = 10 feet, and so on.

3. Total Distance Traveled:

The total distance traveled by the ball includes both the upward and downward motions.

• For each bounce, the ball travels up and then back down. The up and down distances are equal for each bounce.

To calculate the total distance, we can break it into two parts:

• First part: The initial drop (80 feet).
• Second part: The up-and-down bounces.

Distance from the bounces:

For the bounces, we need to sum both the upward and downward paths.

Each bounce height is halved, so the distance traveled during the up-and-down motion of each bounce is twice the height of that bounce. This creates the following sequence for the total bounce distance:

• First bounce: 40 feet up, 40 feet down → Total = 80 feet
• Second bounce: 20 feet up, 20 feet down → Total = 40 feet
• Third bounce: 10 feet up, 10 feet down → Total = 20 feet
• ...
• Seventh bounce: The ball reaches a height of $\frac{80}{2^7} = \frac{80}{128} = 0.625$ feet on the seventh bounce.

The total distance after catching the ball at the top of the seventh bounce means we do not include the final downward motion of the seventh bounce.

4. Summing the Distances:

• The initial drop is 80 feet.
• The total bounce distances is the sum of the geometric series:

$S = 80 + 80\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{64}\right)$

This is a geometric series where the first term $a = 80$ and the common ratio $r = \frac{1}{2}$, summed over 6 terms (since after the seventh bounce, the ball only goes up and is caught at the peak).

The sum of a geometric series is given by:

$S = a \times \frac{1 - r^n}{1 - r}$

where $a = 80$, $r = \frac{1}{2}$, and $n = 6$.

Let me calculate this sum.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the calculations if you'd like to do them manually or I can try again later.

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Here are some related questions that might interest you:

1. How can we apply geometric sequences to other real-world scenarios?
2. What is the sum of an infinite geometric series?
3. How do we calculate the total distance if the ball kept bouncing forever?
4. What role do geometric progressions play in financial models?
5. Can we model population growth using geometric sequences?

Tip: Understanding geometric sequences is crucial for topics like finance, physics, and engineering, where repeated proportional changes occur.