Question

There is a cube in which one pair of opposite faces is painted Black; the second pair of opposite faces is painted Pink and the third pair of opposite faces is painted Brown. This cube is now cut into 216 smaller but identical cubes.How many small cubes are there with at least two different colours
on their faces.​

Answer 1

Answer:

72

Step-by-step explanation:

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Answer 2

• The total number of small cubes with at least two different colours on their faces are equal to 56 .

Given :-

• There is a cube in which one pair of opposite faces is painted Black .
• The second pair of opposite faces is painted Pink .
• The third pair of opposite faces is painted Brown .
• This cube is now cut into 216 smaller but identical cubes .

To Find :-

• How many small cubes are there with at least two different colours on their faces ?

Formula used :-

• Number of cubes with 2 sides painted = 12(n - 2)
• Number of cubes with 3 sides painted = 8 (always)
• n = side of bigger cube .
• Volume of cube = (side)³

Solution :-

Let,

→ side of each smaller cube = 1 unit

so,

→ volume of each smaller cube = (side)³ = (1)³ = 1 unit³

then,

→ volume of 216 smaller cubes = 216 × 1 = 216 unit³

now, let us assume that, side of bigger cube so formed by joing all smaller cubes is equal to n unit .

So,

→ Volume of bigger cube = volume of 216 smaller cubes

→ (n)³ = 216

→ (n)³ = (6)³

cube root both sides,

→ n = 6 unit .

therefore,

→ Total number of small cubes with 2 sides painted = 12(n - 2)

putting n = 6 we get,

→ Total number of small cubes with 2 sides painted = 12(6 - 2) = 12 × 4 = 48 cubes ------ Equation (1)

and,

→ Total Number of small cubes with 3 sides painted = 8 cubes ------- Equation (2)

hence, adding Equation (1) and Equation (2) we get,

→ Total number of small cubes with at least two different colours on their faces = Total number of small cubes with 2 sides painted + Total Number of small cubes with 3 sides painted

→ Total number of small cubes with at least two different colours on their faces = 48 + 8

→ Total number of small cubes with at least two different colours on their faces = 56 cubes (Ans.)

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