The perimeter of a square `S_(1)` is 12 m more than the perimeter of the square `S_(2)`. If the area of `S_(1)` equals three times the area of `S_(2)` minus 11,, then what is the perimeter of `S_(1)` ?

Options

A. 24 m
B. 32 m
C. 36 m
D. 40 m

Answer

To solve the problem step by step, let's denote the side lengths of the squares \( S_1 \) and \( S_2 \) as \( x \) and \( y \) respectively.

### Step 1: Set up the equations based on the problem statement

1. **Perimeter Relationship**:
The perimeter of square \( S_1 \) is 12 m more than the perimeter of square \( S_2 \).
\[ 4x = 4y + 12 \]

2. **Area Relationship**:
The area of square \( S_1 \) is equal to three times the area of square \( S_2 \) minus 11.
\[ x^2 = 3y^2 - 11 \]

### Step 2: Simplify the first equation

From the first equation, we can simplify it:
\[ 4x - 4y = 12 \implies x - y = 3 \implies y = x - 3 \]
This gives us our first equation (Equation 1):
\[ y = x - 3 \]

### Step 3: Substitute \( y \) in the second equation

Now, substitute \( y \) from Equation 1 into the second equation:
\[ x^2 = 3(x - 3)^2 - 11 \]

### Step 4: Expand and simplify the equation

Expanding \( (x - 3)^2 \):
\[ (x - 3)^2 = x^2 - 6x + 9 \]
Now substituting back:
\[ x^2 = 3(x^2 - 6x + 9) - 11 \]
Expanding the right side:
\[ x^2 = 3x^2 - 18x + 27 - 11 \]
This simplifies to:
\[ x^2 = 3x^2 - 18x + 16 \]

### Step 5: Rearranging the equation

Rearranging gives:
\[ 0 = 3x^2 - x^2 - 18x + 16 \]
\[ 0 = 2x^2 - 18x + 16 \]

### Step 6: Solve the quadratic equation

Now, we can solve the quadratic equation:
\[ 2x^2 - 18x + 16 = 0 \]
Dividing the entire equation by 2:
\[ x^2 - 9x + 8 = 0 \]
Now, we can factor this:
\[ (x - 8)(x - 1) = 0 \]
Thus, \( x = 8 \) or \( x = 1 \).

### Step 7: Find corresponding \( y \) values

Using \( y = x - 3 \):
- If \( x = 8 \), then \( y = 8 - 3 = 5 \).
- If \( x = 1 \), then \( y = 1 - 3 = -2 \) (not valid since side lengths cannot be negative).

### Step 8: Calculate the perimeter of \( S_1 \)

Now, we only consider \( x = 8 \):
\[ \text{Perimeter of } S_1 = 4x = 4 \times 8 = 32 \text{ m} \]

### Final Answer
The perimeter of square \( S_1 \) is **32 m**.