in a square field of side x+2y there is a circular path of radius x+y Calculate the area of the remaining field and also find the cost of levelling the land at rate of x2 +2x-4

Given the problem:

1. The side of the square field is \( x + 2y \), so its area is the square of the side:
   \[
   A_{\text{square}} = (x + 2y)^2 = (x + 2y)(x + 2y)
   = x^2 + 4xy + 4y^2
   \]

2. The radius of the circular path is \( x + y \), so the area of the circular path is:
   \[
   A_{\text{circle}} = \pi (x + y)^2 = \pi (x^2 + 2xy + y^2)
   \]

3. The remaining area of the field (the area of the square minus the area of the circle) is:
   \[
   A_{\text{remaining}} = A_{\text{square}} - A_{\text{circle}} = (x^2 + 4xy + 4y^2) - \pi(x^2 + 2xy + y^2)
   \]
   Simplifying:
   \[
   A_{\text{remaining}} = (x^2 + 4xy + 4y^2) - \pi(x^2 + 2xy + y^2)
   \]
   \[
   A_{\text{remaining}} = (1 - \pi)x^2 + (4 - 2\pi)xy + (4 - \pi)y^2
   \]

Now, to find the **cost of leveling the remaining land**:

The cost is given by the rate \( x^2 + 2x - 4 \), so the total cost for the remaining land is:
\[
\text{Cost} = \left( x^2 + 2x - 4 \right) \times A_{\text{remaining}}
\]
Substituting the expression for \( A_{\text{remaining}} \):
\[
\text{Cost} = (x^2 + 2x - 4) \times \left( (1 - \pi)x^2 + (4 - 2\pi)xy + (4 - \pi)y^2 \right)
\]

This expression represents the total cost of leveling the land.

Would you like further details on the cost expansion, or help with any specific part of the problem?

### Follow-up questions:
1. What happens if the rate of leveling the land changes?
2. How does the cost change if \( x \) and \( y \) are equal?
3. What is the area when the square field has side 10 and the circular path has radius 7?
4. Can we calculate the cost for a specific value of \( x \) and \( y \)?
5. How would the calculation differ for a non-square field?

### Tip:
When solving area problems, visualize the geometry to better understand the relationships between shapes.

In a square field of side x+2y there is a circular path of radius x+y. Calculate the area of the remaining field and also find the cost of leveling the land at rate of x^2 + 2x - 4.

Learn how to calculate the remaining area of a square field with a circular path, and find the cost of leveling the land at a given rate. This solution covers concepts like area calculations, polynomial expansions, and cost analysis, suitable for students in Grades 10-12.

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