If the area of an equilateral triangle is given as 900 m 2 , then what is its perimeter?

Question

Examoogle · Accepted Answer

The correct answer is Option A: $60 \sqrt[4]{27} 	ext{ m}$. Calculate $a^2$ first, then find $a = \sqrt{\frac{3600}{\sqrt{3}}} = 60 	imes (3)^{-1/4}$. The perimeter $3a$ becomes $180 	imes 3^{-1/4}$, which simplifies using $3^{1} 	imes 3^{-1/4} = 3^{3/4}$ or manipulation to match the $60 	imes \sqrt[4]{27}$ form.

Step-by-Step Solution & Concepts:
Shortcut Trick: Calculate $a^2$ first, then find $a = \sqrt{\frac{3600}{\sqrt{3}}} = 60 	imes (3)^{-1/4}$. The perimeter $3a$ becomes $180 	imes 3^{-1/4}$, which simplifies using $3^{1} 	imes 3^{-1/4} = 3^{3/4}$ or manipulation to match the $60 	imes \sqrt[4]{27}$ form.

Formula Used: $$Area = \frac{\sqrt{3}}{4} a^2$$, $$Perimeter = 3a$$

Given: Area of an equilateral triangle = 900 m^2.

Steps: Equate area to formula: The formula for the area of an equilateral triangle is $A = \frac{\sqrt{3}}{4} a^2$. Given $A = 900$, we write: $900 = \frac{\sqrt{3}}{4} a^2$.. Solve for side 'a': To find $a$, take the square root of both sides: $a = \sqrt{\frac{3600}{\sqrt{3}}} = \frac{60}{3^{1/4}}$.. Calculate perimeter: Perimeter $P = 3a$. Substituting the value of $a$, $P = 3 	imes 60 	imes 3^{-1/4} = 60 	imes 3^1 	imes 3^{-1/4} = 60 	imes 3^{3/4}$. Since $3^3 = 27$, this is $60 \sqrt[4]{27}$.

Answer: A is correct because the perimeter of the triangle with area 900 m^2 is calculated to be 60 √[4]{27} m.

Exam Tip: This type of simplification frequently appears in SSC/Railway exams involving surds and indices; always simplify the base to the smallest prime factor (here 3) before calculating powers.

Examoogle · Answer

Option B: $60 \sqrt[4]{81} 	ext{ m}$

Examoogle · Answer

Option C: $60 \sqrt[3]{27} 	ext{ m}$

Examoogle · Answer

Option D: $60 \sqrt[4]{21} 	ext{ m}$

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