Equilateral Triangle Area Formula

The area ($A$) of an equilateral triangle with side length ($s$) is given by the formula:

$A = \frac{\sqrt{3}}{4} s^2$

Calculating Side Length from Area

We are given the area $A = 900 \text{ m}^2$. We can use the area formula to find the side length ($s$):

1. Substitute the area into the formula:

   $900 = \frac{\sqrt{3}}{4} s^2$

2. Rearrange the formula to solve for $s^2$:

   $s^2 = \frac{900 \times 4}{\sqrt{3}} = \frac{3600}{\sqrt{3}}$

3. Rationalize the denominator:

   $s^2 = \frac{3600 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{3600 \sqrt{3}}{3} = 1200 \sqrt{3}$

4. Solve for $s$ by taking the square root:

   $s = \sqrt{1200 \sqrt{3}} = \sqrt{1200} \times \sqrt{\sqrt{3}}$

   $s = \sqrt{400 \times 3} \times (3^{1/2})^{1/2} = 20\sqrt{3} \times 3^{1/4}$

   $s = 20 \times 3^{1/2} \times 3^{1/4} = 20 \times 3^{(1/2 + 1/4)} = 20 \times 3^{3/4}$

   $s = 20 \times \sqrt[4]{3^3} = 20 \sqrt[4]{27} \text{ m}$

Calculating Perimeter

The perimeter ($P$) of an equilateral triangle is $P = 3s$. Substitute the calculated side length:

$P = 3 \times (20 \sqrt[4]{27})$

$P = 60 \sqrt[4]{27} \text{ m}$

Final Answer

The calculated perimeter matches Option A.