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Three cubes of metal whose edges are in the ratio 3: 4: 5 are melted down into a single cube whose diagonal is $23$ cm. What are the edges of three cubes?

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6 cm, 8cm, 10cm

6 cm, 9cm, 11cm

61cm, 8cm, 10cm

6 cm, 18cm, 10cm

answer is A.

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Detailed Solution

Given- the edges of the three cubes are in the ratio

3 : 4 : 5

and the diagonal of the new cube is

123 c m

.
Let’s assume that the edges be 3x, 4x, and 5x.
The volume of the cube is given by,

V = s i d e 3

.
Therefore, the volume of the resulting cube,

⇒ V = 3 x 3 + 4 x 3 + 5 x 3 ⇒ V = 27 x 3 + 64 x 3 + 125 x 3 ⇒ V = 216 x 3

Let’s assume that

l

is the edge of the resulting cube, then, the diagonal of the cube is

3 l

⇒ 123 = 3 l

⇒ l = 12 c m

The volume of the cube

V = s i d e 3

Therefore,

V = l 3

⇒ 216 x 3 = 123 ⇒ 6 x 3 = 123 ⇒ 6 x = 12 ⇒ x = 2

Hence, the edges of the three cubes are,

3 x = 3 (2) ⇒ 3 x = 6 c m 4 x = 4 (2) ⇒ 4 x = 8 c m 5 x = 5 (2) ⇒ 5 x = 10 c m

Thus, the edges of the three cubes are,

and

.
Hence, option 1 is the correct option.

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