Question : 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?
Option 1: 4 : 7
Option 2: 7 : 8
Option 3: 7 : 12
Option 4: 2 : 3
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Correct Answer: 7 : 8
Solution :
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.
We know that,
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 = a
3
The lateral surface area of cube = 4 × a
2
Where a = side of the cube
According to the question,
lbh = 8 × a
3
⇒ 18 × 36 × 72 = 8 × a
3
⇒ a
3
= 18 × 36 × 9
⇒ a = $\sqrt[3]{9 × 2 × 9 × 2 × 2 × 9}$
⇒ a = 18 cm
Also,
2(lb + bh + hl) : 8 × 4 × a
2
⇒ 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
Hence, the correct answer is 7 : 8.
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