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April 24th, 2014, 11:55 AM  #1 
Newbie Joined: Apr 2014 From: Mexico Posts: 2 Thanks: 1  If I know the optimal proportions of a shape, why are they not the same in reality?
So, my math homework asked me to find the relation between the height and the radius of a cylinder can so that the surface area is minimized and the volume is maximized. If I'm not mistaken, this proportion should be h/r=2 for any can whose form is a cylinder with a top and a bottom. The question I have is more of a practical question. If the optimal proportion is known, how come in real life none of the metal cylinder cans you find at the store (whether they are tuna cans, dog food cans, vegetables cans, etc...) have these proportions? Is there a mathematical way to prove that these cans (the ones you find at the store) are also efficiently designed, even though they don't have the required proportions? Thanks! 
April 24th, 2014, 02:13 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Good question! I think that the answer is that the mathematical solution optimizes for volume only, but there are many other considerations to take into account. For example: * Marketing: How big will it look? Can it be distinguished from other similar products? * Production: How hard is it to make? * Transportation: Can it stack? Is it stable enough to not tip over? etc. 
April 24th, 2014, 02:39 PM  #3 
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, rodochoa! While taking Calculus, I had the same thoughts. It seems that $\displaystyle h = 2r$ makes for a rather homely can. Code: *       *       h     *       * 2r Very few canned items have that proportion (condensed milk, some brands of mushrooms, cans of mixed nuts, Crisco?) I assume that the companies have testmarketed various shapes to see what appeals to consumers. The familiar Campbell's soup cans have: $\displaystyle h \approx 3.76r$ and tunafish cans have: $\displaystyle h \approx 0.74r$. 
April 24th, 2014, 04:50 PM  #4 
Newbie Joined: Apr 2014 From: Mexico Posts: 2 Thanks: 1 
Thanks! I also thought about it being a consequence of the manufacturers desire to stand out and create a more visually appealing product. However, does this mean that, at least from the mathematical point of view, those cans are nor efficiently designed?

April 24th, 2014, 05:03 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  It means that they use more material than a container optimized only for surface area. But I'd be hesitant to call them inefficient, because I think that (for the most part) the shapes are good, in that the deviations serve other purposes.

April 24th, 2014, 06:15 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra 
One important consideration would be how easy the can is to hold while you apply a can opener. Another might be that they want to minimise the radius so to minimise the surface area of the product that is exposed to the air when the can is opened. This makes it easier to preserve the contents for a few hours/days after opening.


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derivative, optimal, optimization problems, proportions, reality, shape 
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