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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.