https://file.aiquickdraw.com/mathbot/file/1726507417177w94hzio6.jpg

Let's solve the problem step by step.

### Given:
- Initial height \( h_1 = 40 \, \text{m} \)
- The ball rebounds to 60% of its previous height after each bounce, i.e., the rebound ratio \( r = 0.6 \).

#### Part (a): Maximum height after the fifth bounce

The height after each bounce follows a geometric sequence where:
- \( h_1 = 40 \, \text{m} \)
- Common ratio \( r = 0.6 \)

The formula for the \( n \)-th term of a geometric sequence is:
\[
h_n = h_1 \times r^{n-1}
\]
For the fifth bounce:
\[
h_5 = 40 \times 0.6^{5-1} = 40 \times 0.6^4
\]
Let me calculate this.The maximum height the ball reaches after hitting the ground for the fifth time is approximately **5.18 meters**.

#### Part (b): Total vertical distance traveled when the ball hits the ground for the eighth time

For this, we need to calculate the total distance, which includes:
1. The initial drop of 40 m.
2. The sum of the distances for each bounce up and down until the eighth time.

After the first drop:
- The ball rises to 60% of the previous height, then falls the same distance, repeating this for each bounce.

The total distance for the first 8 ground hits is the sum of a geometric series:
\[
\text{Total distance} = 40 + 2 \times \left( \sum_{n=1}^{7} h_1 \times r^n \right)
\]
I'll calculate this now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still guide you on how to solve this.

For part (b), you can use the following steps:

1. **First drop:** The ball falls 40 m.
2. **Rebound distances:** The ball rebounds to 60% of its height each time.
3. **Total vertical distance traveled up to the eighth ground hit** will be:
   \[
   \text{Total distance} = 40 + 2 \times \sum_{n=1}^{7} (40 \times 0.6^n)
   \]

This sum is geometric, so you can calculate each bounce height and sum them accordingly.

#### Part (c): Total vertical distance before the ball comes to rest

The ball’s motion follows a geometric series where:
\[
\text{Total distance} = 40 + 2 \times \left( \frac{40 \times 0.6}{1 - 0.6} \right)
\]
This formula calculates the infinite sum of the geometric series for all the bounces. If you calculate it manually, it should give the total vertical distance before the ball comes to rest.

Would you like me to explain any specific parts or help further? Here are some related questions to think about:

1. What is the total distance traveled after 4 bounces?
2. How would the result change if the rebound ratio was 0.7 instead of 0.6?
3. What is the formula for the sum of an infinite geometric series, and how does it apply here?
4. How many bounces would the ball make before reaching less than 1 meter in height?
5. How would the total distance differ if the initial drop height was doubled?

**Tip:** The geometric series sum formula is useful for solving real-world problems involving repeated reductions or growth!

A ball is dropped from a height of 40 m and rebounds to 60% of the distance it previously fell. a) Determine the maximum height the ball reaches after hitting the ground the fifth time. b) Determine the total vertical distance traveled when the ball hits the ground for the eighth time. c) Determine the total vertical distance traveled before the ball comes to rest.

Learn how to calculate the height of a ball after five bounces and the total distance traveled after eight bounces using geometric sequences. This problem also covers the total distance traveled before the ball comes to rest. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation