Would you like a cup or cone with that?

On Sunday, in an effort to save the planet, I rode my bike to the Suncoast to make yet another losing bet on no safety, as well as a couple other props. When I was ready to leave I noticed a water bottle and cone-shaped cups near where I parked my bike in the valet area. So I took a cup and had a drink. While drinking I got to wondering if forming the cups in the shape of a cone was the most efficient use of paper. I've certainly seem the disposable paper cups in a near cylinder shape as well. I say "near" because they are wider at the top, to allow stacking one inside another.

So, I spent hours on finding a calculus solution to the optimal dimensions of the cone-shaped cup, to maximize volume given a fixed surface area. In that end I threw up my hands in frustration. It is funny that the cylinder is so easy, but when you introduce the triangulation of a cone it becomes an unholy mess.

So I cheated and used the "goal seek" function in Excel. Using that method, I find that the cone-shaped cup is 7.46% more efficient than a cylinder shape. In other words, given the same amount of paper, an optimal-sized cone will hold 7.46% more volume than an optimal-sized cylinder.

Here are some more details, based on 1 unit of paper for both cups.

Cone
radius = 0.428691379
hypotenuse = 0.742515249 (distance from tip to any point on the base)
height= 0.606261162
volume= 0.116675015

Cylinder
radius=0.325735008
height=0.325735008
volume=0.108578336

No particular question here. I just thought I'd share with you how I spent my morning.

The paper no doubt went into the cone-forming machine as a rectangle. It is well known in engineering that it is difficult to shape a rectangle into such shapes, wasting paper. Has this been factored in here?

I remember doing a problem similar to this in Calculus, but I don't think it was the same problem. I may have to dig out the calc book tomorrow night and try to find it...

Thanks. I saw this immediately before I left work. I spend the 20 minute ride home thinking about this - sans radio!

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When you consider the paper required, DO NOT unroll the cone, and draw a rectangle around of it. You're not going to manufacture only one cone.

The manufacturer uses paper on a wide roll, thousands of feet long. The paper is cut in a pattern that lays out the shape in the most efficient way possible. I.E. The scrap paper is kept to a minimum.

The volume differences *may* make the cone a better choice, but there are several other factors:

1 - The cone uses a single cut of paper. The cup uses two. Therefore the cone may be cheaper in raw materials and/or manufacturing equipment, etc.

2 - The cone stacks more efficiently, allowing more cones in the dispenser, and less need to refill it.

3 - Cones can't be put down with water in them. This discourages people from taking an excess of water.

4 - Because of the limited use, the cones are less likely to be stolen.

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Now that we know why cones are the container of choice for water coolers, is the specific height to radius used the most efficient?

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On a related note, the "Cylinder" cup is not a true cylinder. Any geometry people out there that can identify it's geometric shape?

Quote: DJTeddyBear

Because of the limited use, the cones are less likely to be stolen.

Interesting point, I never thought of that.

Quote: DJTeddyBear

Now that we know why cones are the container of choice for water coolers, is the specific height to radius used the most efficient?

Good question. I threw mine away but will investigate the next time I get one. Once I checked this on various sizes of cans, and indeed found they were close to optimal, sometimes a little off. This is why the small cans tend to be taller, by the way, than a big number-10 can, the greater the volume, the more efficient it is to make the can shorter and wider.

If anyone else has a paper cone cup can you tell us any two of the three: diameter, height, distance from tip to edge.

Quote:

On a related note, the "Cylinder" cup is not a true cylinder. Any geometry people out there that can identify it's geometric shape?

I would call it a truncated cone, but there may be another term for it.

Quote: DJTeddyBear

3 - Cones can't be put down with water in them. This discourages people from taking an excess of water.

In a workplace, you also go through a lot more, because you don't save a cup and take it back to your desk. People use a lot more in a day.

Quote: DJTeddyBear

On a related note, the "Cylinder" cup is not a true cylinder. Any geometry people out there that can identify it's geometric shape?

Isn't it a frustum?

Being a high school calc teacher, I felt the need to try the calculus solution. It works out pretty nicely. The exact radius is (A/(3Pi^2))^(1/4), where A is the fixed area of the paper.

Quote: DJTeddyBear

Now that we know why cones are the container of choice for water coolers, is the specific height to radius used the most efficient?

For the record, the cone certainly seems to be very close to the shape required for maximum volume. But I doubt that's the entire reason for the specific height/radius ratio.

If it was much more slender, it wouldn't stack as compactly, and if it was much more oblique, it would be difficult to grip and remove from the dispenser.

Quote: cclub79

In a workplace, you also go through a lot more, because you don't save a cup and take it back to your desk. People use a lot more in a day.

I think in the workplace, people are more likely to use a mug or water bottle.

Quote: drebbin37

Isn't it a frustum?

For the record, I had no idea. I'm glad someone answered before people badgered me for asking without knowing the answer.

Confirmation: http://en.wikipedia.org/wiki/Frustum

Quote: Wikipedia

In geometry, a frustum (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it.

Lots of interesting info and formulas on that page. Interesting stuff and formulas on the http://en.wikipedia.org/wiki/Cone_(geometry) page, too.

Quote: drebbin37

The exact radius is (A/(3Pi^2))^(1/4), where A is the fixed area of the paper.

Oops, that A should have been squared!

In case anyone is interested, I just broke this question into three questions and posted them on my site mathproblems.info, problems 208 to 210.