If the area of an equilateral triangle is `x` and its perimeter is `y`, then which one of the following is correct?

Options

A. `y^(4) = 432x^(2)`
B. `y^(4) = 216x^(2)`
C. `y^(2) = 432x^(2)`
D. None of these

Answer

To solve the problem, we need to establish the relationship between the area \( x \) and the perimeter \( y \) of an equilateral triangle.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]

2. **Set up the equations**:
- Given that the area is \( x \) and the perimeter is \( y \), we can write:
\[ x = \frac{\sqrt{3}}{4} a^2 \quad \text{(1)} \]
\[ y = 3a \quad \text{(2)} \]

3. **Express \( a \) in terms of \( y \)**:
- From equation (2), we can express \( a \):
\[ a = \frac{y}{3} \]

4. **Substitute \( a \) into the area equation**:
- Substitute \( a = \frac{y}{3} \) into equation (1):
\[ x = \frac{\sqrt{3}}{4} \left(\frac{y}{3}\right)^2 \]

5. **Simplify the equation**:
- Calculate \( \left(\frac{y}{3}\right)^2 \):
\[ \left(\frac{y}{3}\right)^2 = \frac{y^2}{9} \]
- Substitute this back into the equation for \( x \):
\[ x = \frac{\sqrt{3}}{4} \cdot \frac{y^2}{9} \]
- Simplify further:
\[ x = \frac{\sqrt{3} y^2}{36} \]

6. **Rearrange the equation**:
- Rearranging gives us:
\[ y^2 = 36x/\sqrt{3} \]

7. **Final expression**:
- This can be expressed as:
\[ y^2 = 12\sqrt{3}x \]

### Conclusion:
The relationship derived from the area and perimeter of the equilateral triangle is:
\[ y^2 = 12\sqrt{3}x \]