A bouncing tennis ball rebounds each time to a height equal to one half the height of the previous bounce. If it is dropped from a height of 16 m , find the total distance it has travelled when it hits ground for the 10th time :

Options

A. `47(15)/(16)`
B. `37(5)/(16)`
C. `67(11)/(16)`
D. none of these

Answer

To solve the problem of the bouncing tennis ball, we need to calculate the total distance traveled by the ball when it hits the ground for the 10th time. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Bouncing Pattern
The ball is dropped from a height of 16 meters. Each time it bounces, it reaches a height that is half of the previous height. Therefore, the heights of the bounces can be represented as follows:
- 1st drop: 16 m (to the ground)
- 1st bounce: 8 m (up)
- 2nd drop: 8 m (to the ground)
- 2nd bounce: 4 m (up)
- 3rd drop: 4 m (to the ground)
- 3rd bounce: 2 m (up)
- 4th drop: 2 m (to the ground)
- 4th bounce: 1 m (up)
- 5th drop: 1 m (to the ground)
- 5th bounce: 0.5 m (up)
- 6th drop: 0.5 m (to the ground)
- 6th bounce: 0.25 m (up)
- 7th drop: 0.25 m (to the ground)
- 7th bounce: 0.125 m (up)
- 8th drop: 0.125 m (to the ground)
- 8th bounce: 0.0625 m (up)
- 9th drop: 0.0625 m (to the ground)
- 9th bounce: 0.03125 m (up)
- 10th drop: 0.03125 m (to the ground)

### Step 2: Calculate the Total Distance
The total distance traveled by the ball consists of the distance fallen and the distance risen.

1. **First drop**: 16 m (down)
2. **Bounces and drops**: For each bounce and subsequent drop, the distance can be calculated as follows:
- 1st bounce and drop: 8 m up + 8 m down = 16 m
- 2nd bounce and drop: 4 m up + 4 m down = 8 m
- 3rd bounce and drop: 2 m up + 2 m down = 4 m
- 4th bounce and drop: 1 m up + 1 m down = 2 m
- 5th bounce and drop: 0.5 m up + 0.5 m down = 1 m
- 6th bounce and drop: 0.25 m up + 0.25 m down = 0.5 m
- 7th bounce and drop: 0.125 m up + 0.125 m down = 0.25 m
- 8th bounce and drop: 0.0625 m up + 0.0625 m down = 0.125 m
- 9th bounce and drop: 0.03125 m up + 0.03125 m down = 0.0625 m

### Step 3: Summing Up the Distances
Now, we can sum up all the distances:
- Total distance = 16 m (first drop) + (8 + 8) + (4 + 4) + (2 + 2) + (1 + 1) + (0.5 + 0.5) + (0.25 + 0.25) + (0.125 + 0.125) + (0.0625 + 0.0625)

This can be simplified as:
- Total distance = 16 + 2 * (8 + 4 + 2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625)

### Step 4: Calculate the Series
The series inside the parentheses is a geometric series where:
- First term (a) = 8
- Common ratio (r) = 0.5
- Number of terms (n) = 8

The sum of a geometric series can be calculated using the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]

Plugging in the values:
\[ S_8 = 8 \frac{1 - (0.5)^8}{1 - 0.5} = 8 \cdot (1 - \frac{1}{256}) \cdot 2 = 16 \cdot \left(1 - \frac{1}{256}\right) = 16 \cdot \frac{255}{256} = \frac{4080}{256} \]

### Step 5: Final Calculation
Now, adding the first drop:
- Total distance = 16 + 2 * \(\frac{4080}{256}\) = 16 + \(\frac{8160}{256}\) = \(\frac{4096 + 8160}{256} = \frac{12256}{256} = 48.0\)

### Conclusion
The total distance traveled by the tennis ball when it hits the ground for the 10th time is **48 meters**.