A solid metallic cube of side is $6 \sqrt[2]{4}$ cm, is melted and recast into a cuboid of length 12 cm and breadth 9 cm. What is the length (in cm) of the longest diagonal of the cuboid?

17

We know that,

Volume of cube = (side)³

Volume of cuboid = length × breadth × height

Length of diagonal of cuboid = \(\sqrt { l^2 + b^2 + h^2}\)

We have,

Side of cube =  $6 \sqrt[2]{4}$ cm

Length of cuboid = 12 cm

Breadth of cuboid = 9 cm

Let diagonal = d

= Therefore, Volume of cube = Volume of cuboid

= ($6 \sqrt[2]{4}$ )³ = 12 × 9 × Height

= Height = \(\frac{6×6×6×4}{12×9}\) = 8 cm 

= d = \(\sqrt { 12^2 + 9^2 + 8^2}\) = \(\sqrt { 144+81+64 }\) = 17 cm