This problem involves summing up the total vertical distance traveled by the ball, which bounces infinitely, diminishing by a factor of $\frac{2}{3}$ after each bounce.

Step 1: Breaking the problem into parts

1. Initial drop: The ball falls from 13 ft.

   • Distance = $13$ ft.

2. First bounce up: It rebounds $\frac{2}{3}$ of the height.

   • Distance = $13 \times \frac{2}{3}$.

3. First bounce down: The ball falls back down the same distance.

   • Distance = $13 \times \frac{2}{3}$.

   This pattern continues, forming an infinite geometric series for the bounces.

Step 2: Total distance formula

The total distance can be computed as: $\text{Total Distance} = \text{Initial Drop} + 2 \times \text{Sum of all bounces up/down}.$

The series for bounces up/down is geometric: $\text{Sum of bounces} = 13 \times \frac{2}{3} + 13 \times \left(\frac{2}{3}\right)^2 + 13 \times \left(\frac{2}{3}\right)^3 + \dots$

This is an infinite geometric series with:

• First term $a = 13 \times \frac{2}{3}$,
• Common ratio $r = \frac{2}{3}$.

The sum of an infinite geometric series is: $\text{Sum} = \frac{a}{1 - r}.$

Step 3: Calculating the total distance

1. Compute the sum of bounces: