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  4. Three solid cubes have a face diagonal of 4 √2 cm each. Three other solid cubes have a face diagonal of 8 √2 cm each. All the cubes are melted together to form a cube. Find the side of the cube formed (in cm).

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Q. 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. Find the side of the cube formed (in cm).

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Solution:

Side of each of the first three cubes $=\frac{4 \sqrt{2}}{\sqrt{2}}=4 \,cm$.
Side of each of the other three cubes $=\frac{8 \sqrt{2}}{\sqrt{2}}=8 \,cm$.
Let the side of the cube formed be $a cm$.
Total volume of the six cubes $=3\left(4^3+8^3\right)=3(64+512)=1728\, cm ^3$
$\therefore a^3=1728$
$a=12$.