January 21, 2013 7 comments
No pantry is complete without an ample supply of canned goods. In addition to the variety of food that exists in can form, canned foods tend to be less expensive and have a longer shelf life than their uncanned counterparts. It’s no surprise, then, that canned food sales go up when the economy goes down. But are we really saving as much as we could? Or are canned food companies passing on unnecessary cost?
When you go to the supermarket, you’ve probably noticed that soup cans tend to be taller than they are wide. The opposite is true for other products, like canned tuna. Still other cans have a shape somewhere in between. Let’s try to find out if one of these shapes is better than the others!
One way to determine a “best” shape is to find the one that stores the most food using the least amount of material. If manufacturers are able to minimize the amount of material used, it may result in a cheaper product. So, let’s say we have a fixed volume of food that we want to store in a cylindrical can using the smallest amount of material. In other words, we want to make the surface area small. To do this, we need to understand the relationship between volume and surface area.
You may remember that a cylinder with radius r and height h has volume (V) and surface area (S) given by the formulas
V = πr2h,
S = 2πr2 + 2πrh.
(For some discussion on where these formulas come from, see the comments below.)
If the volume is constant, then we can solve for the height in terms of it: h = V/πr2. In turn, we get S = 2πr2 + 2V/r. In other words, the surface area is a function of the radius!
This function is a little hard to wrangle, though, especially without Calculus. The problem is that if you want a small surface area, the radius can’t be too big because 2πr2 is large when r is large. But it also can’t be too small, because 2V/r is large when r is small.
Thankfully, there’s a value of r which is just right – you can see it from the graph below when V = 30. (Click through to explore how S changes as V changes!)
It turns out that regardless of the volume, the surface area is smallest when the can’s diameter equals its height. Of the cans up above, though, only one has this property. The others could hold the same amount using less material. So why don’t they?
Well, this isn’t necessarily a manufacturing failure; it may be a failure of our model. For example, the top of the can is often the last piece attached, and sometimes is even manufactured in a different location. Consequently, it may cost a different amount. If the top is c times as expensive per square inch as the rest of the can, then as c increases, the cheapest design has a smaller and smaller radius. This makes intuitive sense – if the top is expensive, we should make it smaller. (The orange graph represents the cost if the top were twice as expensive as the rest of the can, assuming the other materials cost $1/in.2; as before, you can click through to explore what happens when c and V vary.)
Of course, this is just one possible factor affecting a can’s shape. There are undoubtedly others. For example, one factor may be psychological: taller, thinner cans may appear larger than shorter, rounder cans. Would you believe that the blue and red cylinders actually have the same volume? (Click through to play around with these cylinders; you can adjust the radii and heights, but their volumes will remain identical.)
There are undoubtedly other possible factors as well. But you folks are smart – I’m sure you can come up with alternative explanations on your own!
Teachers: interested in discussing other explanations with your students? Then check out our new lesson, Canalysis!