In reaching her destination, a backpacker walks an average velocity of 1.34 m/s, due west. This average velocity results because she hikes for 6.44 km with an average velocity of 2.68 m/s, due west, turns around, and hikes with an average velocity of 0.447 m/s, due east. How far east did she walk ?

Options

A. 3.5 km
B. 1.8 km
C. 2.4 km
D. 0.81 km

Answer

To solve the problem step by step, we need to analyze the backpacker's journey and calculate how far she walked east after initially walking west.

### Step 1: Understand the given information
- Average velocity of the entire journey = 1.34 m/s due west.
- Distance walked west = 6.44 km = 6440 m.
- Average velocity while walking west = 2.68 m/s.
- Average velocity while walking east = 0.447 m/s.
- Let the distance walked east be \( d \).

### Step 2: Calculate the time taken to walk west
Using the formula for time, \( \text{time} = \frac{\text{distance}}{\text{velocity}} \):
\[ t_1 = \frac{6440 \, \text{m}}{2.68 \, \text{m/s}} \approx 2407.46 \, \text{s} \]

### Step 3: Express the total displacement
The total displacement \( S \) can be expressed as:
\[ S = \text{Distance west} - \text{Distance east} = 6440 \, \text{m} - d \]

### Step 4: Relate total displacement to average velocity
The average velocity \( V_{avg} \) is given by:
\[ V_{avg} = \frac{S}{T} \]
where \( T \) is the total time taken for the entire journey. The total time \( T \) can be expressed as:
\[ T = t_1 + t_2 \]
where \( t_2 = \frac{d}{0.447} \) is the time taken to walk east.

### Step 5: Set up the equation using average velocity
Substituting the expressions for displacement and total time into the average velocity equation:
\[ 1.34 = \frac{6440 - d}{t_1 + t_2} \]
Substituting \( t_1 \) and \( t_2 \):
\[ 1.34 = \frac{6440 - d}{2407.46 + \frac{d}{0.447}} \]

### Step 6: Solve for \( d \)
Cross-multiplying gives:
\[ 1.34 \left( 2407.46 + \frac{d}{0.447} \right) = 6440 - d \]
Expanding and rearranging:
\[ 3214.00 + \frac{1.34d}{0.447} + d = 6440 \]
\[ \frac{1.34d}{0.447} + d = 6440 - 3214.00 \]
\[ \frac{1.34d}{0.447} + d = 3226 \]
Factoring out \( d \):
\[ d \left( \frac{1.34}{0.447} + 1 \right) = 3226 \]
Calculating \( \frac{1.34}{0.447} \):
\[ \frac{1.34}{0.447} \approx 2.997 \]
So,
\[ d (2.997 + 1) = 3226 \]
\[ d \cdot 3.997 = 3226 \]
\[ d = \frac{3226}{3.997} \approx 807.52 \, \text{m} \]

### Step 7: Convert to kilometers
To convert meters to kilometers:
\[ d \approx 0.80752 \, \text{km} \approx 0.81 \, \text{km} \]

### Final Answer
The distance walked east by the backpacker is approximately **0.81 km**.

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