To calculate the total area of the garden plot defined by the poles at $(0,0)$, $(6,0)$, $(4,5)$, and $(2,4)$, we can use the shoelace formula, which calculates the area of a polygon given its vertices in order.

Shoelace Formula

If the vertices of the polygon are $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$, then the area is:

$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|$

Vertices

The given vertices are: $(0,0), (6,0), (4,5), (2,4)$ We will arrange them in order and repeat the first vertex at the end: $(0,0), (6,0), (4,5), (2,4), (0,0)$

Applying the Formula

We calculate each term in the formula step by step:

1. $x_1y_2 = 0 \cdot 0 = 0$
2. $x_2y_3 = 6 \cdot 5 = 30$
3. $x_3y_4 = 4 \cdot 4 = 16$
4. $x_4y_1 = 2 \cdot 0 = 0$

Sum of these terms: $0 + 30 + 16 + 0 = 46$.

Next, calculate the other diagonal terms:

1. $y_1x_2 = 0 \cdot 6 = 0$
2. $y_2x_3 = 0 \cdot 4 = 0$
3. $y_3x_4 = 5 \cdot 2 = 10$
4. $y_4x_1 = 4 \cdot 0 = 0$

Sum of these terms: $0 + 0 + 10 + 0 = 10$.

Subtract and Divide

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

Final Answer

The total area of the garden plot is:

$\boxed{18 \, \text{square feet}}$

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Would you like a detailed explanation of any step or further examples? Here are related questions:

1. What is the shoelace formula and how is it derived?
2. How do you determine the order of vertices for such problems?
3. Can you use this method for non-convex polygons?
4. What are practical applications of the shoelace formula?
5. What are alternative methods to calculate the area of a polygon?

Tip: Always ensure the vertices are ordered sequentially (clockwise or counterclockwise) before applying the shoelace formula!