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A solid metal cube with edge length 6 is melted down and recast, with no loss of metal, into a number of identical smaller solid cubes each with edge length 2. By what factor does the total surface area of all the metal increase compared with the surface area of the original cube?
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Correct answer
B
Two ideas combine. Volume is conserved when the metal is melted, so the number of small cubes is the original volume divided by a small cube's volume: 6³ = 216 and 2³ = 8 give 216/8 = 27 small cubes. The trap is to assume surface area is conserved the same way, but it is not.
The original cube has surface area 6 × 6² = 216. Each small cube has surface area 6 × 2² = 24, and 27 of them total 27 × 24 = 648. The factor of increase is 648/216 = 3. So the answer is B.
Cutting a solid into pieces always increases total surface area because new internal faces get exposed, so a factor of 1 (choice A) is impossible. Choice C, 9, squares the linear scale factor instead of tracking the pieces; choice D, 27, reports the count of cubes; choice E, 54, just doubles that count.
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