To determine the height of the ball at the top of bounce 6, we will analyze the given data step-by-step:
### Step 1: Understand the Pattern of Height Reduction
First, we are given the heights of the ball at the top of five consecutive bounces:
- Bounce 1: 250 feet
- Bounce 2: 200 feet
- Bounce 3: 160 feet
- Bounce 4: 128 feet
- Bounce 5: 102.4 feet
We observe that the height decreases with each bounce. We need to identify a pattern in the height reduction.
### Step 2: Calculate the Ratios Between Consecutive Bounces
We calculate the ratio of the height of each bounce to the height of the previous bounce:
1. Ratio from bounce 1 to bounce 2: [tex]\( \frac{200}{250} = 0.8 \)[/tex]
2. Ratio from bounce 2 to bounce 3: [tex]\( \frac{160}{200} = 0.8 \)[/tex]
3. Ratio from bounce 3 to bounce 4: [tex]\( \frac{128}{160} = 0.8 \)[/tex]
4. Ratio from bounce 4 to bounce 5: [tex]\( \frac{102.4}{128} = 0.8 \)[/tex]
### Step 3: Determine the Average Ratio
We observe that the ratio remains constant at each step, which is 0.8. This means the ball's height is reducing by the same factor of 0.8 with every bounce.
### Step 4: Calculate the Height at Bounce 6
Given that the height at bounce 5 is 102.4 feet, we use the ratio to find the height at bounce 6:
[tex]\[ \text{Height at bounce 6} = \text{Height at bounce 5} \times \text{Ratio} \][/tex]
[tex]\[ \text{Height at bounce 6} = 102.4 \times 0.8 \][/tex]
### Step 5: Perform the Multiplication
[tex]\[ \text{Height at bounce 6} = 81.92 \text{ feet} \][/tex]
Thus, the height of the ball at the top of bounce 6 is 81.92 feet.
