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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 clairière, available under a Creative Commons Attribution-NonCommercialShareAlike 2.0 Generic License.
Is the shape of a soda can truly optimal?
For an on-line presentation, see
http://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 optimization
problems.
Setting the Stage.
Better use of resources is a win-win proposition for economics and for the environment. With the
goal 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, so
we should be able to find the optimal shape that uses the least amount of material.
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of NSF.
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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 21.6563in3 and minimal surface
area (21.6563in3 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 for h, get A as a function of r and find the critical points.
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Because r is the only critical point and A(r) goes to infinity as r →0 and as r → ∞, this must be
the 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 not
need 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 the
sides. 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 cans
that 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, if
you 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, the
thickness 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.) Thicknesses
of 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, ctop and cbot instead 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 off
at 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 tapering
at 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 r and the line of sight distance
s from the observer to the horizon.
Therefore,
All other quantities needed on the preceding pages can be derived from s.
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 this
is 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 then
one 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 problem
done?" and my answer was “That does not matter. Your answer is right."
© 2011 Bernd Schroeder. All rights reserved. Copies may be downloaded from
www.EngageEngineering.org. This material may be reproduced for educational purposes.
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