A square lawn has a 2-m-wide path surrounding it. If the area of the path is 136 m,find the area of the lawn.

Answer

To find the area of the square lawn surrounded by a 2-meter-wide path, we can follow these steps:

### Step 1: Define the variables
Let the side length of the square lawn be \( x \) meters.

### Step 2: Calculate the dimensions of the larger square
Since there is a 2-meter-wide path surrounding the lawn, the dimensions of the larger square (which includes the path) will be:
\[ \text{Side length of larger square} = x + 4 \text{ meters} \]
(This is because the path adds 2 meters to each side of the lawn.)

### Step 3: Calculate the area of the larger square
The area of the larger square (including the path) is given by:
\[ \text{Area of larger square} = (x + 4)^2 \]

### Step 4: Calculate the area of the lawn
The area of the square lawn is:
\[ \text{Area of lawn} = x^2 \]

### Step 5: Calculate the area of the path
The area of the path is the area of the larger square minus the area of the lawn:
\[ \text{Area of path} = \text{Area of larger square} - \text{Area of lawn} \]
Substituting the areas we calculated:
\[ \text{Area of path} = (x + 4)^2 - x^2 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x + 4)^2 = x^2 + 8x + 16 \]
Thus,
\[ \text{Area of path} = (x^2 + 8x + 16) - x^2 = 8x + 16 \]

### Step 7: Set up the equation
We know the area of the path is given as 136 m²:
\[ 8x + 16 = 136 \]

### Step 8: Solve for \( x \)
Subtract 16 from both sides:
\[ 8x = 136 - 16 \]
\[ 8x = 120 \]
Now, divide both sides by 8:
\[ x = \frac{120}{8} = 15 \text{ meters} \]

### Step 9: Calculate the area of the lawn
Now that we have \( x \), we can find the area of the lawn:
\[ \text{Area of lawn} = x^2 = 15^2 = 225 \text{ m}^2 \]

### Final Answer
The area of the lawn is \( 225 \text{ m}^2 \).

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