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A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes? 

A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes?  Correct Answer 7 : 8

Given:

A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm

where l = 18 cm, b = 36 cm and h = 72 cm

It is melted and recast into 8 cubes of the same volume.

Concept used:

The volume of cuboid = lbh

The total surface area of the cuboid = 2(lb + bh + hl)

Where l = length, b = breadth and h = height

The volume of cube = a3

The lateral surface area of cube = 4 × a2

Where a = side of cube

Explanation:

according to the question,

lbh = 8 × a3

⇒ 18 × 36 × 72 = 8 × a3

⇒ a3 = 18 × 36 × 9

⇒ a = √(9 × 2 × 9 × 2 × 2 × 9)

⇒ a = 18 cm

now,

according to the question,

2(lb + bh + hl) : 8 × 4 × a2

⇒ 2(18 × 36 + 36 × 72 + 72 × 18) : 8 × 4 × (18)2

⇒ 2 × 18 × 36(1 + 4 + 2) : 8 × 4 × 18 × 18

⇒ 36 × 36 × 7 : 32 × 18 × 18

⇒ 7 : 8

∴ The ratio is 7 : 8.

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