A square lawn has a path 2m wide around it. The area of the path is 196 sq. m. Find the length of the side of the lawn.

Answer

To solve the problem, we need to find the length of the side of the square lawn given that there is a 2-meter-wide path around it and the area of the path is 196 square meters.

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

### Step 2: Determine the Dimensions of the Larger Square
Since there is a path that is 2 meters wide around the lawn, the dimensions of the larger square (which includes the lawn and the path) will be:
- Length of the side of the larger square = \( x + 2 + 2 = x + 4 \) meters.

### Step 3: Calculate the Area of the Larger Square
The area of the larger square can be calculated as:
\[ \text{Area of larger square} = (x + 4)^2 \]

### Step 4: Calculate the Area of the Lawn
The area of the lawn (the smaller square) 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:
\[ \text{Area of path} = (x + 4)^2 - x^2 \]
We know from the problem statement that the area of the path is 196 sq. m, so we can set up the equation:
\[ (x + 4)^2 - x^2 = 196 \]

### Step 6: Expand the Equation
Expanding the left side:
\[ (x^2 + 8x + 16) - x^2 = 196 \]
This simplifies to:
\[ 8x + 16 = 196 \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ 8x = 196 - 16 \]
\[ 8x = 180 \]
\[ x = \frac{180}{8} = 22.5 \]

### Step 8: Conclusion
Thus, the length of the side of the lawn is \( 22.5 \) meters.