Question

Three cubes of a metal whose edges are in the ratio 3 : 4 : 5 are melted and converted into a single cube whose diagonal is 12$\sqrt{3}$ cm. Find the edges of the three cubes.

Answer 1

Let the edge of the metal cubes be 3x, 4x and 5x.

Let the edge of the single cube be a.

As,

Diagonal of the single cube = 12√3 cm

⇒ $a\sqrt{3}$= $12\sqrt{3}$

⇒ a = 12 cm

Now,

Volume of the single cube = Sum of the volumes of the metallic cubes

⇒ $a^3=(3x{)}^3+(4x{)}^3+(5x{)}^3$

⇒ ${12}^3=27x^3+64x^3+125x^3$

⇒ $1728=216x^3$

⇒ $x^3=\frac{1728}{216}$
⇒ $x^3=8$
⇒ $x^3=2^3$
⇒ x = 2

So, the edges of the cubes are 3 x 2 = 6 cm, 4 x 2 = 8 cm and 5 x 2 = 10 cm.

Hence, the edges of the given three metallic cubes are 6 cm, 8 cm and 10 cm.