Question

If $l$ is the length of a diagonal of a cube of volume $v$, then

• A. $3\sqrt{3v}=l^3$
• B. $3v=l^3$
• C. $3\sqrt{3v}=2l^3$
• D. $\sqrt{3v}=l^3$

Answer 1

The correct option is A $3\sqrt{3v}=l^3$

Given:
$l$ is the length of a diagonal of a cube of volume $v$

Let the side of cube be $a$

$\Rightarrow$ Length of diagonal of cube
$=\sqrt{3}\left(side\right)=\sqrt{3}a$

$\therefore$ Volume of cube $\left(v\right)=a^3$

$\Rightarrow v=a^3$

$\Rightarrow v^{\frac{1}{3}}=a$

$\Rightarrow$ Length of diagonal of cube
$=\sqrt{3}{v}^{\frac{1}{3}}$

$\Rightarrow l=\sqrt{3}{v}^{\frac{1}{3}}$

On cubing both sides, we get $l^3=3\sqrt{3}v$

Hence, (d) is the correct option.