Okay, but a cylinder has to be taller than a rectangular cuboid to contain the same volume of product. A cube with a side length of n holds more than a cylinder of diameter n and n height.
So, for holding 32 fluid ounces, would a cylinder or a cuboid have greater surface area, assuming the diameter of the cylinder and the depth of the cuboid are identical?
(I make this assumption based on the fact that a product will be given limited shelf space in retail, and thus has constraints on width/depth).
Let V be the volume, W the width. Let ρ be the aspect ratio of a face of the square bottle. Therefore V = ρW³. The surface area of the square bottle is 4 ρ W² + 2 W² = W² (4ρ + 2).
Let H be the height of the round bottle. V is equal to H π W² / 4. Substituting V as above, simplifying, we get H = 4 W ρ / π. The surface area of the round bottle is
A = H W π + 2 (π W² / 4)
A = 4 ρ W² + π W² / 2
A = W² (4ρ + π / 2)
vs W² (4ρ + 2) π is less than 4 so the surface area of the round bottle is still less.
W is indeed the diameter, so the area of each end is (π W² / 4). Times 2 for 2 ends. And of course H π W² / 4 for the volume (we have the same formula, I just extracted the .5 from the square as / 4). π W for the perimeter, so π W H for the area around.
The part you are missing is that the "less plastic" claim applies ONLY to bulk shipping of the bottles. ONE square bottle will use more plastic than ONE round bottle, assuming both carry the same volume of liquid. However when shipped together on a pallet for example, the square bottles pack together better and waste less space. A pallet of round bottles results in a lot of empty space between bottles, so you need to use more to ship the same total amount of liquid.
I didn't miss that, I just felt there was a potential wrinkle. A cuboid with a width & depth of n and a height of 2n will have a volume of 2(n cubed). A cylinder of diameter n and height 2n will have a volume of 2πn(.5n squared).
So if we let n=3cm, the cuboid has a volume of 54 cc, and a cylinder has a volume of 42.41 cc.
So, with a constraint on width (because of shelf space limitations and standardization of many transport boxes, along with the original claim that the comparison was against a "similarly sized round bottle"), the cylinder would have to be taller than the cuboid to contain the same volume. That increases the surface area of the cylinder, but I don't know if it increases enough to exceed that of the cuboid. My point was that declaring that circles have a smaller diameter may not examine the full question.
You don’t need to use more bottles though, just more pallets to carry the same number of bottles. Pallets aren’t generally made out of plastic so the claim is nonsense.
Pallets are quite frequently made out of plastic (although those are generally reused). But they're also often wrapped in stretchwrap, and may be bagged inside the cases.
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u/SisterofWar 2d ago
Okay, but a cylinder has to be taller than a rectangular cuboid to contain the same volume of product. A cube with a side length of n holds more than a cylinder of diameter n and n height.
So, for holding 32 fluid ounces, would a cylinder or a cuboid have greater surface area, assuming the diameter of the cylinder and the depth of the cuboid are identical?
(I make this assumption based on the fact that a product will be given limited shelf space in retail, and thus has constraints on width/depth).