A 210 cm³ block of metal was melted to make a solid tetrahedron of height 4√3 cm. The remaining metal was poured into a cuboid mould of dimensions 6 cm × 5 cm. What is the height of the cuboid so formed? 

Given:

A 210 cm³ block of metal was melted to make a solid tetrahedron of height 4√3 cm

Dimensions of cuboid = 6 × 5 × height

Formula used:

Height of tetrahedron = √6/3 × side

Volume of tetrahedron = √2/12 × (sides)³

Volume of cuboid = length × breadth × height

Calculation:

As we know, a tetrahedron is a triangular pyramid with four equilateral triangles as its four faces 

Let the side of the regular tetrahedron be a cm 

∵ Height of tetrahedron = √6/3 × side

⇒ 4√3 = √6/3 × side

⇒ side = 6√2 cm 

Also, Volume of tetrahedron = √2/12 × (sides)³

⇒ Volume of tetrahedron = √2/12 × (6√2)³ = 72 cm³

∵ Volume of metal = 210 cm³

⇒ Volume of remaining metal = 210 – 72 = 138 cm³

But, the remaining metal was poured in a cuboid mould of dimensions 6 cm × 5 cm,