If the width of the path around a square field is 4.5 m and the area of the path is 252 `m^(2)`, then the length of the side of the field is:

Options

A. 9.5m
B. 9 m
C. 8 m
D. 8.5 m

Answer

To find the length of the side of the square field, we can follow these steps:

### Step 1: Define Variables
Let the length of the side of the inner square (the field) be \( x \) meters. The width of the path around the field is given as 4.5 meters.

### Step 2: Calculate the Side Length of the Outer Square
The outer square, which includes the path, will have a side length of:
\[ x + 2 \times 4.5 = x + 9 \text{ meters} \]

### Step 3: Calculate the Area of the Outer Square
The area of the outer square can be expressed as:
\[ \text{Area of outer square} = (x + 9)^2 \]

### Step 4: Calculate the Area of the Inner Square
The area of the inner square (the field) is:
\[ \text{Area of inner square} = x^2 \]

### Step 5: Calculate the Area of the Path
The area of the path is the difference between the area of the outer square and the area of the inner square:
\[ \text{Area of path} = (x + 9)^2 - x^2 \]
We know from the problem statement that the area of the path is 252 m². Therefore, we can set up the equation:
\[ (x + 9)^2 - x^2 = 252 \]

### Step 6: Expand and Simplify the Equation
Expanding the left side:
\[ (x^2 + 18x + 81) - x^2 = 252 \]
This simplifies to:
\[ 18x + 81 = 252 \]

### Step 7: Solve for \( x \)
Now, we can isolate \( x \):
\[ 18x = 252 - 81 \]
\[ 18x = 171 \]
\[ x = \frac{171}{18} = 9.5 \text{ meters} \]

### Conclusion
The length of the side of the field is \( 9.5 \) meters.

---