The problem involves a ball being dropped from a height of 18 feet, bouncing back to a height that is $\frac{1}{3}$ of the previous height each time. We need to calculate three quantities:

1. The total vertical distance covered by the ball in its downward movements.
2. The total vertical distance covered by the ball in its upward movements.
3. The total vertical distance covered by the ball (both upwards and downwards).

Step-by-Step Solution

1. Vertical Distance in Descending (Downward) Movements

The descending distances are:

• First fall: 18 feet
• Second fall: $\frac{1}{3} \times 18 = 6$ feet
• Third fall: $\frac{1}{3} \times 6 = 2$ feet
• Fourth fall: $\frac{1}{3} \times 2 = \frac{2}{3}$ feet
• And so on.

This is an infinite geometric series with the first term $a = 18$ and the common ratio $r = \frac{1}{3}$.

The sum $S_d$ of an infinite geometric series is: $S_d = \frac{a}{1 - r} = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 27 \text{ feet}$

2. Vertical Distance in Ascending (Upward) Movements

The ball bounces up after each fall, except the first drop. Each bounce height follows a geometric sequence:

• First bounce up: 6 feet
• Second bounce up: 2 feet
• Third bounce up: $\frac{2}{3}$ feet
• And so on.

This is also an infinite geometric series with the first term $a = 6$ and common ratio $r = \frac{1}{3}$.

The sum $S_u$ of this infinite series is: $S_u = \frac{a}{1 - r} = \frac{6}{1 - \frac{1}{3}} = \frac{6}{\frac{2}{3}} = 9 \text{ feet}$

3. Total Vertical Distance Covered by the Ball

The total vertical distance $S_{\text{total}}$ is the sum of the descending and ascending distances: $S_{\text{total}} = S_d + S_u = 27 + 9 = 36 \text{ feet}$

Answers

1. The distance (in feet) vertically in descent covered by the ball is 27 feet.
2. The distance (in feet) vertically in ascent covered by the ball is 9 feet.
3. The total vertical distance (in feet) covered by the ball is 36 feet.

Would you like more details on any part of this solution, or do you have further questions?

Related Questions

1. What would happen if the rebound height ratio were $\frac{1}{2}$ instead of $\frac{1}{3}$?
2. How would the solution change if the ball started from a different height?
3. What if the ball bounced indefinitely? How does convergence play a role in this problem?
4. Can this geometric series model be applied to other real-world scenarios?
5. How can we calculate the time of each bounce if we know the acceleration due to gravity?

Tip

In problems involving repetitive geometric decreases, geometric series are often the best approach to find total sums for both infinite and finite cases.