https://file.aiquickdraw.com/mathbot/file/1726508352357pc7slqvy.jpg

Let's solve the problem step by step based on the image provided. 

### Given:
- The ball is dropped from a height of **40 m**.
- The ball rebounds to **60%** of the height it fell from each time.

### Part a: 
**Determine the maximum height the ball reaches after hitting the ground the fifth time.**

The ball reaches **60%** of the height it fell from each time, so this forms a geometric sequence. The initial height is **40 m**, and the common ratio \( r \) is **0.60**.

The height reached after hitting the ground for the fifth time is:
\[
h_5 = h_1 \times r^4
\]
Where:
- \( h_1 = 40 \, \text{m} \) is the initial height.
- \( r = 0.60 \).
- \( 4 \) because after the fifth bounce, the ball has rebounded 4 times.

Let's calculate:
\[
h_5 = 40 \times (0.60)^4
\]

### Part b:
**Determine the total vertical distance travelled when the ball hits the ground for the eighth time.**

For this, the total distance is the sum of both the distance falling down and the rebounds up. The distance after each fall and rebound follows a geometric series.

1. The ball falls **40 m** on the first fall.
2. Then it rebounds to **60%** of the height and falls back again.

So the total distance is:
\[
\text{Total Distance} = 40 + 2 \times \left( 40 \times 0.60 + 40 \times 0.60^2 + \dots + 40 \times 0.60^7 \right)
\]
Here, the series follows:
\[
\sum_{n=1}^{7} 40 \times 0.60^n
\]

### Part c:
**Determine the total vertical distance travelled before the ball comes to rest.**

This requires calculating the sum of an infinite geometric series. The total distance the ball travels before coming to rest is:
\[
\text{Total Distance} = 40 + 2 \times \sum_{n=1}^{\infty} 40 \times 0.60^n
\]

The sum of an infinite geometric series is given by:
\[
S = \frac{a}{1 - r}
\]
Where:
- \( a = 40 \times 0.60 \) (the first rebound height),
- \( r = 0.60 \).

Now, I'll compute the values for parts a, b, and c.### Solutions:

**Part a:**
The maximum height the ball reaches after hitting the ground for the fifth time is approximately **5.18 m**.

**Part b:**
The total vertical distance travelled when the ball hits the ground for the eighth time is approximately **156.64 m**.

**Part c:**
The total vertical distance travelled before the ball comes to rest is **160.0 m**.

Let me know if you would like more detailed explanations or have any questions!

Here are some related questions for further practice:

1. What would be the total vertical distance after the ball hits the ground for the 10th time?
2. If the rebound ratio were 70%, how would the results change?
3. How would you calculate the distance if the initial height was doubled?
4. What happens to the total distance if the ball only rebounds 50% of the height each time?
5. Can you find the total distance if the ball bounces infinitely with different rebound ratios?

**Tip:** The sum of an infinite geometric series can be very useful in physics and finance, where repetitive processes tend to converge over time!

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

Explore how to calculate the heights and distances a ball travels in a geometric series when it rebounds to a fraction of its previous height. This problem demonstrates the application of geometric series in physics, including the formulas for the height after multiple bounces and the total distance traveled.

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