Question : A cuboid whose sides are 8 cm, 27 cm, and 64 cm is melted to form a new cube. What is the respective ratio of the total surface area of the cuboid and cube?
Option 1: 307 : 216
Option 2: 291 : 203
Option 3: 349 : 248
Option 4: 329 : 237
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Correct Answer: 307 : 216
Solution : Sides of cuboid = 8 cm, 27 cm, and 64 cm Let $a$ be the side of the cube. The volume of the cuboid = volume of the cube $lbh$ = $a^3$ 8 × 27 × 64 = $ a^3$ $a$ = $\sqrt[3]{8 × 27 × 64}$ = 2 × 3 × 4 = 24 cm Total surface area of cuboid = $2(lb+ bh + hl)$ Total surface area of cuboid = 2 × (8 × 27 + 27 × 64 + 64 × 8) = 2 × (216 + 1728 + 512) = 2 × 2456 = 4912 The total surface area of the cube = $6a^2$ = 6 × 24 × 24 = 3456 Required ratio = 4912 : 3456 = 307 : 216 Hence, the correct answer is 307 : 216.
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