A metallic rectangular cuboid of dimensions is melted and some more metal is added to make it a cube, the measure of whose side is an integer. What is the minimum volume of metal to be added and what is the side?

Given: Dimensions of cuboid are 4cm $\times$ 6cm $\times$ 8cm.

It is melted and some ore is added to make it cube ,

To do: To find the minimum volume of metal to be added and at what side?

Solution:

The volume of the cuboid = Length x Width x Height

So, volume of given cuboid = 4 x$\times$6 $\times$ 8 = 192 

It is given that the cuboid is made into a cube.

The volume of the cube = $(side)^{3}$

It is given that some minimum volume is added to cuboid to make a cube whose side is an integer.

If we look at the perfect cube integers, the cube nearest to 192 is 216.

So, minimum volume of metal to be added = 216 - 192 = 24 cm³

Also, Volume of cube = $(side)^{3}$ = 216

So side = ∛216 = 6 cm

So, the minimum volume of metal to be added is 24 cm^3 and the side of the cube

is 6 cm.