Question

In Figure 2, $d$ is the distance from vertex $A$ to vertex $B$. Find the volume of the cube in terms of $d$.

• A. $9d^3\sqrt{3}$
• B. $\frac{d^3\sqrt{3}}{9}$
• C. $d^3\sqrt{3}$
• D. $\frac{d^3\sqrt{3}}{3}$

Answer 1

The correct option is B $\frac{d^3\sqrt{3}}{9}$

Let the length of side of cube be ${}^{\prime }{a}^{\prime }$.

Length of the body diagonal is ${(a^2+a^2+a^2)}^{1/2}=\sqrt{3}a$.

It is given that body diagonal is ${}^{\prime }{d}^{\prime }$.

$\Rightarrow d=\sqrt{3}a\Rightarrow a=\frac{d\sqrt{3}}{3}$

Now the volume of the cube is $a^3$

As $a$ is $d\sqrt{3}$, Volume becomes ${\left(\frac{d\sqrt{3}}{3}\right)}^3$

$\Rightarrow V=\left(\frac{3\sqrt{3}{d}^3}{27}\right)$

$=\frac{d^3\sqrt{3}}{9}$