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This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Anyopinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do n

necessarily reflect the views of NSF.

Using Everyday Examples in Engineering (E3)

Optimization Problems: Is the Shape of a Soda Can Truly Optimal?

Bernd Schroeder

Louisiana Tech University

http://www2.latech.edu/~schroder/

Photo Credit: Cans by Emmental de clairire, available under a Creative Commons Attribution-NonCommercial-

ShareAlike 2.0 Generic License.

Is the shape of a soda can truly optimal?

For an on-line presentation, seehttp://www2.latech.edu/~schroder/videos/SVC/diffform_opt2.wmv

Where it Fits.

Presentation can be weaved into any calculus I or business calculus class on optimizationproblems.

Setting the Stage.Better use of resources is a win-win proposition for economics and for the environment. With thegoal of using as little aluminum as possible, the shape of soda cans was redesigned in the 1980s,

when soda cans were given their current shape. A soda can is roughly a right circular cylinder, sowe should be able to find the optimal shape that uses the least amount of material.

www.EngageEngineering.org

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First Attempt.The amount of material used in a soda can is related to the surface area of the can. So let's first

try this problem.

Find the dimensions of the right circular cylinder with volume 21.6563in3and minimal surface

area (21.6563in

3

is 12 fl.oz. in cubic inches)

We need the formulas

They should be obtained from the visualization of a right circular cylinder.

Solve the volume forh, getA as a function ofrand find the critical points.

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Because ris the only critical point andA(r) goes to infinity as r0 and as r, this must bethe radius of the can with the smallest surface area. The height is

But that looks more like a cube!

Now, everyone loves a good conspiracy theory (and sanity takes a leave of absence when that

happens). It would not be logical for industry to spend money (on aluminum) that it does notneed to spend.

So what went wrong here? The can with the smallest surface area need not be the cheapest can.

Check a regular soda can and you will find that the top and the bottom are much sturdier than thesides. The top has the pull tab, so it must have a certain thickness so that opening the can does

not tear the metal in an inappropriate way. The bottom must have a certain thickness so that cansthat lay horizontally on top of each other (crates sometimes end up sideways in shipping, even if

only for a brief moment) do not crush each other.

The minimum thickness of the sides is dictated by the manufacturing process. Soda cans are, ifyou will, cups at first. Then soda is filled in and the lid is pressed on with a force of around 250

pounds. This force causes the metal in can and lid to fuse and make a tight seal. Thus, thethickness of the sides is dictated by the fact that the can must withstand at least a downward

force of 250 pounds. (Yes, with good balance, you can stand on an empty soda can.) Thicknessesof bottom and sides are similarly dictated by strength requirements so that cans cannot easily be

crushed by sideways impact with other cans during packing and shipping.

So let's try this again with actual industry parameters. The parameters are the costs"(thicknesses) of top, bottom and sides.

Find the cheapest can" with V= 21.6563in3, csides= 0.0041in, ctop= 0.0093in and

cbot= 0.0113in

We're going to use V,csides, ctopand cbotinstead of the decimals. (Actual specs for the

thicknesses.)

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These measurements are pretty close to the actual height of 4.812in and radius of 1.354in. Note

that an actual soda can is not a right circular cylinder. It's a cylinder in the middle that tapers offat the ends. The actual shape was computed using numerical analysis (the finite element method)

to obtain the desired strength parameters as well as lowest possible use of materials. The taperingat the top is for materials savings, as the lid is thicker than the tapered sides. The design of the

bottom is for stacking and strength.

Shorter derivation of the formula, without using the chain rule.The eyes of the observer, the point on the horizon, and the center of the Earth make a right

triangle with hypotenuse of length r+ h and the other sides being rand the line of sight distances from the observer to the horizon.

Therefore,

All other quantities needed on the preceding pages can be derived froms.

This is still calculus (using a good understanding of tangent lines).It's also a teachable moment that could be presented at the end of class (simpler alternative

method that works in a special case vs. general approach that works in other cases, too) or if thisis done in group work if students figure it out. That's how I got the formula: I gave the open

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ended problem as group work in class, students started working, drew the right pictures and thenone group after another came up with the formula above. Class was shorter than planned and a

lot of fun, and we could still talk about satellite networks, etc.

The most important point to drive home, especially with current concerns about test obsession

and mindless mimicking, was this. A student asked me Is this how you want the problemdone?" and my answer was That does not matter. Your answer is right."

2011 Bernd Schroeder. All rights reserved. Copies may be downloaded fromwww.EngageEngineering.org. This material may be reproduced for educational purposes.