If the volume of a cube is $ 512 ext{ cm}^3 $, then the measure of the main diagonal of the cube will be:

Question

If the volume of a cube is $ 512 	ext{ cm}^3 $, then the measure of the main diagonal of the cube will be:

Examoogle · Accepted Answer

The correct answer is Option D: $8\sqrt{3} 	ext{ cm}$. Memorize common perfect cubes like $8^3 = 512$. Once the side $a=8$ is found, the diagonal is simply $a\sqrt{3}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Memorize common perfect cubes like $8^3 = 512$. Once the side $a=8$ is found, the diagonal is simply $a\sqrt{3}$.

Formula Used: $$V = a^3$$, $$d = a\sqrt{3}$$

Given: Volume of the cube V = 512 cm^3

Steps: Calculate the side length: Given the volume of the cube $V = a^3 = 512$, take the cube root to find the side length $a$.. Calculate the main diagonal: The formula for the main diagonal $d$ of a cube with side length $a$ is $d = a\sqrt{3}$. Substitute $a=8$ into the formula.

Answer: D is correct because the side of the cube is $\sqrt[3]{512} = 8 	ext{ cm}$, making the main diagonal $8\sqrt{3} 	ext{ cm}$.

Exam Tip: This concept is frequently tested alongside lateral surface area ($4a^2$) and total surface area ($6a^2$) of a cube in various competitive exams.

Examoogle · Answer

Option A: $17\sqrt{3} 	ext{ cm}$

Examoogle · Answer

Option B: $14\sqrt{3} 	ext{ cm}$

Examoogle · Answer

Option C: $7\sqrt{3} 	ext{ cm}$

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