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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