Three solid cubes have a face diagonal of 4 2 cm each. Three other solid cubes have a face diagonal of 8 2 cm each. All the cubes are melted together to form a big cube. Find the side of the cube formed (in cm).

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Three solid cubes have a face diagonal of $42 c m$ each. Three other solid cubes have a face diagonal of $82 c m$ each. All the cubes are melted together to form a big cube. Find the side of the cube formed (in cm).

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10

12

13

15

answer is B.

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Detailed Solution

Given:
Three solid cubes having an equal length of the face diagonal given by,

⇒ d 1 = 42 cm

Let us assume the length of the side of the square of these three solid cubes be

a 1

.
We know that the face of a cube is a square and the diagonal of the square is

2

times the length of the side of the square.
So, we get, for first three cubes,

⇒ d 1 = 2 × a 1

Substituting

d 1 = 42 cm

we get,

⇒ 42 = 2 × a 1

Solving this, we get,

⇒ a 1 = 4 cm

So, the volume of the three cubes = 3 × volume of one cube

⇒ V 1 = 3 × a 1 3

Substituting

a 1 = 4

we get,

⇒ V 1 = 3 × 43 ​ ⇒ V 1 = 3 × 64 ​ ⇒ V 1 = 192 cm 3

Similarly,
For three other solid cubes having the same length of the face diagonal given by-

⇒ d 2 = 82 cm

Let us assume the length of the side of the square of these three solid cubes be

a 2

.
Then for the three other cubes, the length of the diagonal of the square

⇒ d 2 = 2 × a 2

Substituting

d 2 = 82 cm

we get,

⇒ 82 = 2 × a 2

Solving this, we get,

⇒ a 2 = 8 cm

So, the volume of the three cubes = 3 × volume of one cube

⇒ V 2 = 3 × a 2 3

Substituting

a 2 = 8

we get,

⇒ V 2 = 3 × 83 ​ ⇒ V 2 = 3 × 512 ​ ⇒ V 2 = 1536 cm 3

Now, the total volume of all the six cubes combined

V = V 1 + V 2

Substituting the values,

⇒ V = 192 + 1536 ​ V = 1728 cm 3

Let the side of this big cube be a then, its volume is,

⇒ V = a 3 ⇒ ​ 1728 = a 3 ⇒ a = 12 cm

Therefore, the side of the big cube formed after melting all cubes is

12 c m

and the correct option is 2.

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