In reaching her destination, a backpacker walks with an average velocity of 1 m/s, due west. This average velocity results, because she hikes for 6 km with an average velocity of 3 m/s due west, turns around, and hikes with an average velocity of 0.3 m/s due east.How far east did she walk (in kilometers)?

To determine how far east the backpacker walked, we need to analyze her entire journey step by step, taking into account her velocities and the distances she traveled in both directions. Let's break it down.

Understanding the Journey

The backpacker's journey can be divided into two parts: the hike to the west and the hike back to the east. We know the following:

• First leg: Hiked 6 km due west at an average velocity of 3 m/s.
• Second leg: Hiked back east at an average velocity of 0.3 m/s.
• Overall average velocity for the entire trip: 1 m/s due west.

Calculating the Time for Each Leg

First, let's calculate the time taken for the westward hike:

Using the formula for time, which is:

Time = Distance / Velocity

For the westward leg:

Time (west) = 6 km / 3 m/s

We need to convert kilometers to meters for consistency:

6 km = 6000 m

So:

Time (west) = 6000 m / 3 m/s = 2000 seconds

Next, we need to find the total time for the entire journey. Since we know the average velocity for the entire trip is 1 m/s, we can express the total distance and total time relationship:

Average Velocity = Total Distance / Total Time

Let’s denote the distance traveled east as d (in meters). The total distance traveled is:

Total Distance = 6000 m (west) + d (east)

Now, the total time for the entire journey is:

Total Time = Time (west) + Time (east)

We can express Time (east) as:

Time (east) = d / 0.3 m/s

Setting Up the Equation

Now, substituting these into the average velocity formula:

1 m/s = (6000 m + d) / (2000 s + d / 0.3 m/s)

To simplify, we can multiply both sides by the denominator:

1 m/s * (2000 s + d / 0.3 m/s) = 6000 m + d

This leads to:

2000 s + d / 0.3 m/s = 6000 m + d

Solving for d

Now, let’s multiply through by 0.3 to eliminate the fraction:

0.3 * 2000 s + d = 0.3 * 6000 m + 0.3 * d

Which simplifies to:

600 s + d = 1800 m + 0.3d

Rearranging gives:

600 s = 1800 m - 0.7d

Now, isolate d:

0.7d = 1800 m - 600 s

Convert 600 seconds to meters using the average velocity of 1 m/s:

600 m = 600 s * 1 m/s

Now substituting back:

0.7d = 1800 m - 600 m = 1200 m

Finally, solving for d:

d = 1200 m / 0.7 = 1714.29 m

Converting to Kilometers

To convert meters back to kilometers:

d = 1714.29 m / 1000 = 1.714 km

Final Answer

Thus, the backpacker walked approximately 1.71 kilometers east.