After striking a floor a rubber ball rebounds (7/8)th of the height from which it has fallen. Find the total distance that it travels before coming to rest, if it is gently dropped from a height of 420 meters?

Options

A. 2940
B. 6300
C. 1080
D. 3360

Answer

To solve the problem of finding the total distance traveled by the rubber ball before coming to rest, we can break it down step by step.

### Step 1: Understand the Problem
The ball is dropped from a height of 420 meters and rebounds to (7/8) of the height it fell from. We need to calculate the total distance traveled by the ball until it comes to rest.

### Step 2: Identify the Heights
1. **Initial Drop Height (h)**: The ball is dropped from a height of 420 meters.
2. **Rebound Height**: After hitting the ground, the ball rebounds to (7/8) of the height it fell from. Therefore, the first rebound height is:
\[ \text{Rebound Height} = \frac{7}{8} \times 420 = 367.5 \text{ meters} \]

### Step 3: Calculate Subsequent Rebounds
The ball will continue to rebound to (7/8) of the height of the previous drop. We can find the heights of subsequent rebounds:
- Second Rebound Height:
\[ \text{Second Rebound Height} = \frac{7}{8} \times 367.5 = 320.625 \text{ meters} \]
- Third Rebound Height:
\[ \text{Third Rebound Height} = \frac{7}{8} \times 320.625 = 280.546875 \text{ meters} \]
- This process continues indefinitely.

### Step 4: Formulate the Total Distance
The total distance traveled by the ball consists of the initial drop and the sum of all the rebounds. The distance can be calculated as:
\[ \text{Total Distance} = \text{Initial Drop} + 2 \times (\text{Sum of Rebounds}) \]
The factor of 2 accounts for both the upward and downward travel of the rebounds.

### Step 5: Calculate the Sum of the Rebounds
The heights of the rebounds form a geometric series where:
- First term \( a = 367.5 \)
- Common ratio \( r = \frac{7}{8} \)

The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{367.5}{1 - \frac{7}{8}} = \frac{367.5}{\frac{1}{8}} = 367.5 \times 8 = 2940 \text{ meters} \]

### Step 6: Calculate the Total Distance
Now substituting back into the total distance formula:
\[ \text{Total Distance} = 420 + 2 \times 2940 = 420 + 5880 = 6300 \text{ meters} \]

### Final Answer
The total distance that the rubber ball travels before coming to rest is **6300 meters**.

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