The problem states that a triangular plot of land has dimensions of 220 meters and 100 meters for its two perpendicular sides. Inside this land, a rectangular swimming pool is to be built with one of its vertices on the hypotenuse of the land's triangle. The ratio of the pool's length to its width is 11:5, and we are tasked with finding the area of the pool's base.

Steps:

1. Calculate the area of the triangle: The plot is a right triangle with legs 220 m and 100 m. The area $A_{\text{triangle}}$ is given by: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 220 \times 100 = 11,000 \, \text{m}^2$

2. Calculate the hypotenuse: Using the Pythagorean theorem, the hypotenuse $c$ is: $c = \sqrt{220^2 + 100^2} = \sqrt{48400 + 10000} = \sqrt{58400} \approx 241.62 \, \text{m}$

3. Assign variables to the dimensions of the pool: Let the length of the pool be $11x$ meters and the width be $5x$ meters (based on the given ratio 11:5). The area $A_{\text{pool}}$ of the pool is then: $A_{\text{pool}} = 11x \times 5x = 55x^2$

4. Use the hypotenuse to find $x$: Since one vertex of the pool is on the hypotenuse, the pool’s dimensions must fit within the triangle. The pool's dimensions should be proportional to the triangle, and solving the value of $x$ allows us to determine the area.

Let me calculate the value of $x$ and the final area of the pool.The hypotenuse of the triangular plot of land is approximately 241.66 meters.

Now, to calculate the area of the pool, we can use the given ratio of 11:5. Since the pool's dimensions fit within the triangle, we can assume that the pool's total area is proportional to the area of the triangle.

Next, I will determine the pool's base area based on this information.It seems I misinterpreted part of the proportional scaling earlier. Instead, let's directly reconsider the triangle's dimensions in comparison to the pool's given dimensions of $11:5$. The proportional relationship was unclear.

Let me adjust for the correct approach, using more accurate measurements based