Question

Three solid cubes have a face diagonal of $4\sqrt{2}$ cm each. Three other solid cubes have a face diagonal of $8\sqrt{2}$ cm each. All the cubes are melted together to form a cube. The side of the cube so formed is of length ___________.

Answer 1

We know that each face of a cube is a square.

Let the side of solid cube having face diagonal of $4\sqrt{2}$ cm each be x cm.

Diagonal of a square = $\sqrt{2}\times \mathrm{Side}=\sqrt{2}x$

$\therefore \sqrt{2}x=4\sqrt{2}$

⇒ x = 4 cm

Now, let the side of solid cube having face diagonal of $8\sqrt{2}$ cm each be y cm.

$\therefore \sqrt{2}y=8\sqrt{2}$

⇒ y = 8 cm

Suppose the side of the bigger cube obtained on melting the given cubes be a cm.

∴ Volume of the bigger cube = 3 × Volume of cube having side 4 cm + 3 × Volume of cube having side 8 cm

$$\begin{gathered} \Rightarrow a^3=3\times 4^3+3\times 8^3 \\ \Rightarrow a^3=192+1536=1728 \\ \Rightarrow a^3={\left(12\right)}^3 \\ \Rightarrow a=12\mathrm{cm} \end{gathered}$$

Three solid cubes have a face diagonal of $4\sqrt{2}$ cm each. Three other solid cubes have a face diagonal of $8\sqrt{2}$ cm each. All the cubes are melted together to form a cube. The side of the cube so formed is of length __12 cm__.