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Three cubes of a metal whose edges are in the ratio 3: 4: 5 are melted and converted into a single cube whose diagonal is $123 cm .$ What are the edges of the three cubes?

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4cm, 6cm, 8cm

6cm, 8cm, 10cm

8cm, 10cm, 12cm

6cm, 7cm, 8cm

answer is B.

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Detailed Solution

Given are three cubes with the edges in the ratio 3: 4: 5 and the diagonal of the single cube is

123 cm .

We have the following formula,
Volume of cube

= a 3

Taking the edges of three cubes as 3x, 4x, 5x, then,
Volume of each cube will be,

= (3 x) 3 = 27 x 3 = (4 x) 3 = 64 x 3 = (5 x) 3 = 125 x 3

The sum of their volumes is,

= 27 x 3 + 64 x 3 + 125 x 3 = 216 x 3

The value of x will be,
Volume of new cube = volume of the three cubes

∴ a 3 = 216 x 3 ∴ a = 6 x

We have the diagonal of new cube then,

a 2 + a 2 + a 2 = 123 3 a 2 = 123 3 a = 123 a = 1233 ∴ a = 12

Since

a = 6 x

12 = 6 x x = 2

The edges of three cubes will be,

3 x = 3 × 2 = 6 cm 4 x = 4 × 2 = 8 cm 5 x = 5 × 2 = 10 cm

Thus the edges of three cubes are 6cm, 8cm and 10cm.
Therefore the correct option is 2.

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