The positions of the poles define the vertices of a quadrilateral. To calculate the total area of the garden plot, we can use the shoelace formula, which calculates the area of a polygon given its vertices.

Vertices of the Quadrilateral:

1. $(0,0)$
2. $(6,0)$
3. $(4,5)$
4. $(2,4)$

Shoelace Formula:

If the vertices are $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$, the area is given by:

$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|$

Where $(x_{n+1}, y_{n+1})$ is the first vertex $(x_1, y_1)$ to close the polygon.

Calculation:

Using the vertices in order $(0,0), (6,0), (4,5), (2,4), (0,0)$:

$\text{Area} = \frac{1}{2} \left| (0 \cdot 0 + 6 \cdot 5 + 4 \cdot 4 + 2 \cdot 0) - (0 \cdot 6 + 0 \cdot 4 + 5 \cdot 2 + 4 \cdot 0) \right|$

Simplify:

$\text{Area} = \frac{1}{2} \left| 0 + 30 + 16 + 0 - (0 + 0 + 10 + 0) \right|$

$\text{Area} = \frac{1}{2} \left| 46 - 10 \right| = \frac{1}{2} \cdot 36 = 18 \, \text{square feet}$

Final Answer:

The total area of the garden plot is 18 square feet.

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Would you like a detailed explanation of the formula or further assistance?

Here are some related questions to explore:

1. How does the shoelace formula work for other shapes?
2. What are alternative methods to calculate the area of a polygon?
3. How would the calculation change if the coordinates were in a different unit?
4. What is the importance of the order of vertices in the shoelace formula?
5. Can the shoelace formula handle polygons with more than four sides?

Tip: Always list the vertices in order (clockwise or counterclockwise) when using the shoelace formula to avoid errors.