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The question asks for the work done to move an object along a given path in a vector field. To solve this problem, we need to calculate the work done using the formula for work in a force field, which is:

\[ W = \int \mathbf{F} \cdot d\mathbf{r} \]

where \( \mathbf{F} \) is the force vector, and \( d\mathbf{r} \) is the displacement vector.

**Step 1: Analyzing the Image**
From the image, the object moves along a square path, and the forces involved are probably constant along the path segments.

**Step 2: Determining the Force**
Let's assume the force vector \( \mathbf{F} \) is constant throughout the path, and its components in the \( x \) and \( y \) directions are \( F_x \) and \( F_y \), respectively.

**Step 3: Path Analysis and Calculation**
The path is divided into segments along the grid lines, where we can assume the displacement in the \( x \) and \( y \) directions is uniform along each segment.

1. **Horizontal Segment** (e.g., \( A \to B \)):  
   Displacement \( d\mathbf{r} \) in the \( x \)-direction, with force \( F_x \).

2. **Vertical Segment** (e.g., \( B \to C \)):  
   Displacement \( d\mathbf{r} \) in the \( y \)-direction, with force \( F_y \).

The work done along each segment can be summed up:

- \( W_{AB} = F_x \Delta x \)
- \( W_{BC} = F_y \Delta y \)
- Similarly, calculate for other segments.

**Step 4: Summing the Work Done**

Sum the work done along all segments to get the total work done. Since the image provided is unclear about the specific vector field and forces, I will proceed with the assumption that the work can be calculated as:

\[ W = F_x \Delta x + F_y \Delta y \]

where \( \Delta x \) and \( \Delta y \) are the distances along the \( x \) and \( y \) axes, respectively.

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Let's calculate the work done:

1. If the force is provided or can be assumed from the given choices, we can use that information to determine the work done.

2. The answer choices suggest specific work values, which may correspond to specific forces applied along the path.

Would you like more details on specific calculations or have any particular forces in mind?

Learn how to calculate the work done by a force moving an object along a given path in a vector field. This problem involves analyzing the force components and path segments to determine the total work done. Suitable for advanced high school students studying physics or calculus.

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