A square lawn is surrounded by a path 2.5 m wide. If the area of the path is `165 m^(2)` find the area of the lawn. 

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a square lawn surrounded by a path that is 2.5 meters wide. The area of the path is given as 165 m². We need to find the area of the lawn.

### Step 2: Define Variables
Let the side length of the square lawn be \( x \) meters.

### Step 3: Calculate the Dimensions of the Outer Square
Since the path is 2.5 meters wide on all sides, the dimensions of the outer square (which includes the lawn and the path) will be:
- Length of the outer square = \( x + 2.5 + 2.5 = x + 5 \) meters.

### Step 4: Calculate the Areas
1. **Area of the outer square**:
\[ \text{Area of outer square} = (x + 5)^2 \]
2. **Area of the inner square (lawn)**:
\[ \text{Area of inner square} = x^2 \]

### Step 5: Set Up the Equation
The area of the path is the area of the outer square minus the area of the inner square:
\[ \text{Area of path} = \text{Area of outer square} - \text{Area of inner square} \]
Given that the area of the path is 165 m², we can write:
\[ (x + 5)^2 - x^2 = 165 \]

### Step 6: Simplify the Equation
Expanding the left side:
\[ (x + 5)^2 = x^2 + 10x + 25 \]
So the equation becomes:
\[ x^2 + 10x + 25 - x^2 = 165 \]
This simplifies to:
\[ 10x + 25 = 165 \]

### Step 7: Solve for \( x \)
Subtract 25 from both sides:
\[ 10x = 140 \]
Now, divide by 10:
\[ x = 14 \text{ meters} \]

### Step 8: Calculate the Area of the Lawn
Now that we have the side length of the lawn, we can find its area:
\[ \text{Area of the lawn} = x^2 = 14^2 = 196 \text{ m}^2 \]

### Final Answer
The area of the lawn is **196 m²**.

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