A square field has an area of 400 square meters. Posts are set at all corners of the field. What is the longest distance between any two posts?

Answer

To solve the problem of finding the longest distance between any two posts set at the corners of a square field with an area of 400 square meters, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a square field with posts at each corner. We need to find the longest distance between any two posts.

2. **Draw a Diagram**: Visualize the square field. Label the corners as A, B, C, and D. The posts are located at these corners.

3. **Identify the Length of the Side**: The area of the square field is given as 400 square meters. The formula for the area of a square is:
\[ \text{Area} = \text{side}^2 \]
Therefore, we can find the length of each side (s) by taking the square root of the area:
\[ s = \sqrt{400} = 20 \text{ meters} \]

4. **Calculate the Diagonal**: The longest distance between any two posts will be the diagonal of the square. The formula for the diagonal (d) of a square in terms of its side length is:
\[ d = s \sqrt{2} \]
Substituting the value of s:
\[ d = 20 \sqrt{2} \text{ meters} \]

5. **Conclusion**: The longest distance between any two posts, which are located at opposite corners of the square, is:
\[ \text{Longest distance} = 20 \sqrt{2} \text{ meters} \]

### Final Answer:
The longest distance between any two posts is \( 20 \sqrt{2} \) meters.