An equilateral triangle has area x and perimeter y. Find the correct relation between x and y.

Introduction / Context:
For an equilateral triangle with side a, both area and perimeter have closed forms in terms of a. Eliminating a yields an identity connecting x (area) and y (perimeter). The algebra leads to a simple power relation y^4 = 432 x^2.

Given Data / Assumptions:

• Side length a (equilateral).
• Area x = (√3/4) * a^2.
• Perimeter y = 3a.

Concept / Approach:
Express a in terms of y (a = y/3), substitute into the area formula to get x as a function of y, then isolate y in terms of x and simplify by squaring as needed to clear radicals.

Step-by-Step Derivation:
x = (√3/4) * (y/3)^2 = (√3/4) * y^2 / 9 = (√3/36) * y^2y^2 = (36/√3) * xSquare both sides: y^4 = (36^2 / 3) * x^2 = 1296/3 * x^2 = 432 x^2

Verification / Alternative check:
Pick a specific a (e.g., a = 6): y = 18, x = (√3/4)*36 = 9√3. Compute both sides to confirm equality numerically.

Why Other Options Are Wrong:
216 is half of 432 (from a common algebra slip); y^2 = 432 x^2 mismatches dimensions (units would not balance).

Common Pitfalls:
Dropping or mishandling √3 when eliminating a; forgetting to square both sides to remove the radical properly.

Final Answer:
y4 = 432 x2