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Rectangular Prism Surface Area Versus Volume

Directions: What is least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches?



What kind of rectangular prism maximizes volume and minimizes surface area?



Understanding that the optimal shape is a cube is critical to solving this problem.  With that in mind, each side of the cube has a length of 4 inches.  So the area of one face is 16 square units and the total surface area is 96 square units.

Source: Robert Kaplinsky

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  1. That is the wrong answer the right answer is length to be 4 height to be 8 and with to be 2

    • Robert Kaplinsky

      Hi Holly. I believe that there might be a misunderstanding. A 4 x 8 x 2 rectangular prism has a surface area of 112 square units from these calculations: 2(4 x 8) + 2(4 x 2) + 2(8 x 2).

      A cube has a surface area of 96 square units. The question asks about the least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches. So, it doesn’t appear that the answer is wrong.

      What am I missing?

  2. I like the idea behind the problem, but I’m not crazy about the solution method. If a kid does know that the cube is optimal, it’s not very hard to figure out the dimensions that give 64 cubic inches. It’s a more interesting problem if they DON’T know the cube is optimal (which my kids don’t), but then I think it will be very difficult for them to prove it is.

    What about:

    What are the possible combinations for 3 whole-number dimensions for a rectangular prism with volume 64 cm^3?

    Which of these combinations gives the greatest surface area? the least? the median?

    (If the median is what I think it is, that’s kind of cool and makes me wonder about other cube numbers for volume…)

    I’ll post what I think are the answers in a different comment in case they’re spoilers.

    • OK, here are my answers to my questions. Hope somebody will check my work! Maybe I’ll just give it to my brightest problem-solvers tomorrow!

      Possible dimensions:
      1 x 1 x 64 SA 258 greatest = long skinny
      1 x 2 x 32 SA 196
      1 x 4 x 16 SA 168
      1 x 8 x 8 SA 160 median = flat square [interesting, no?]
      2 x 2 x 16 SA 136
      2 x 4 x 8 SA 112
      4 x 4 x 4 SA 96 least = cube

      (Yeah, I know, units, units. Also, these are literally back of an envelope calculations, so use with caution.)

      • Robert Kaplinsky

        I love all your contributions Julie, especially how you take a theoretically good problem and highlight tweaks to improve it. Could you please submit these new problems for the site?

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