To find the expenditure incurred for soil preparation, we first need to calculate the area of the annular region (the circular strip) where the soil preparation will be done. The formula for the area of an annulus (the region between two concentric circles) is:

$\text{Area} = \pi \times \left( R^2 - r^2 \right)$ where:

• $R$ is the outer radius,
• $r$ is the inner radius.

Given:

• Inner radius $r = 50 \, \text{m}$,
• Outer radius $R = 71 \, \text{m}$,
• Cost of soil preparation = ₹ 2 per $\text{m}^2$.

Step 1: Calculate the area

Substitute the values of $R$ and $r$ into the formula:

$\text{Area} = \pi \times \left( 71^2 - 50^2 \right)$ $\text{Area} = \pi \times \left( 5041 - 2500 \right)$ $\text{Area} = \pi \times 2541$ $\text{Area} \approx 3.1416 \times 2541 \approx 7977.9 \, \text{m}^2$

Step 2: Calculate the expenditure

Now, to find the expenditure, we multiply the area by the cost per square meter:

$\text{Expenditure} = \text{Area} \times \text{Cost per m}^2$ $\text{Expenditure} = 7977.9 \times 2 \approx ₹ 15,955.80$

Final Answer:

The expenditure incurred for soil preparation is approximately ₹ 15,955.80.

Would you like further clarification on any of the steps? Here are some related questions that could expand on this:

1. How do we calculate the area of a circular region?
2. What are the real-world applications of the formula for the area of an annulus?
3. How can we adjust the cost of soil preparation if the radius changes?
4. What would be the impact on the total expenditure if the cost per square meter increased?
5. Can this approach be applied to other types of landscaping or construction projects?

Tip: When dealing with circular areas, always remember that the difference in radii is crucial for annular regions—don't forget to square the radii before subtracting!