To solve this, we can use vector analysis. The hikers travel two legs of the journey, each in a specific direction (true bearings). We need to break these movements into their horizontal (west/east) and vertical (north/south) components and then compute the total westward distance.

Step 1: Convert the bearings into standard Cartesian coordinates (relative to the north).

• First leg (0.7 km at 240° bearing):

  • The bearing of 240° means the direction is 240° clockwise from due north.

  • 240° is 60° south of west, so this is the same as a bearing of 180° + 60°.

  • We can break this movement into its north-south and east-west components.

  • Westward component: $0.7 \times \sin(240^\circ)$

  • Southward component: $0.7 \times \cos(240^\circ)$

• Second leg (1.3 km at 300° bearing):

  • The bearing of 300° is 60° west of north, equivalent to 360° - 300° = 60° west of north.

  • We can break this movement into its north-south and east-west components.

  • Westward component: $1.3 \times \sin(300^\circ)$

  • Southward component: $1.3 \times \cos(300^\circ)$

Step 2: Calculate the westward distance.

We need the sum of the westward components of both legs.

Step 3: Perform the calculations.

Let's calculate:

1. First leg (0.7 km at 240°):

   • $\sin(240^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866$
   • $\cos(240^\circ) = -\frac{1}{2} \approx -0.500$
   • Westward component: $0.7 \times -0.866 = -0.606$ km (westward)

2. Second leg (1.3 km at 300°):

   • $\sin(300^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866$
   • $\cos(300^\circ) = \frac{1}{2} \approx 0.500$
   • Westward component: $1.3 \times -0.866 = -1.127$ km (westward)

Step 4: Total westward distance.

The total westward distance is the sum of the two components:

• $-0.606$ km (first leg) + $-1.127$ km (second leg) = -1.733 km.

Since the direction is westward, the total distance they have traveled westward from their starting point is 1.733 km.

Final Answer:

The hikers have traveled 1.733 km westward.

---

Would you like more details on the solution? Or do you have any questions?

Here are 5 related questions:

1. How do you calculate the eastward component of the displacement vector?
2. What are the differences between bearings and compass directions?
3. How would the solution change if the hikers had traveled on different bearings (e.g., 270° and 330°)?
4. Can you explain the concept of vector addition in navigation problems?
5. How can we apply the same method to solve for the total north-south displacement?

Tip: Always convert bearings to Cartesian angles (relative to north or east) before using trigonometric functions like sine and cosine.