Are Soup And Cereal Companies Evil?

Unfortunately, exposing evil empires will require some knowledge of calculus. If you understand first derivatives, you should be fine. If you don't, I'll try my best to explain what you need to know, but it may prove to be difficult.

To preface, this is one of my favorite things to do in math, because it's a real world problem. Many, many students want to stick to equations and not bother with word problems. Unfortunately, you don't see many equations in real life; you have to make them yourself from information given. While you have all the information you need to solve these problems, they do present a way to apply mathematics to problems we face when grocery shopping.

There are two properties of a container that we care about for this exercise. The first is the container's volume. This is important because we want to know how much soup or cereal our container will hold. The second is the container's surface area. This is important because we want to know how much the container will cost to make, since that depends on how much material we need to make the container.

We'll start with soup companies, since the math is easier.

We know that soup comes in cylindrical cans. The volume of a cylinder is given by the formula pi times radius (r) squared times height (h) (the first formula below). The surface area is a little more complicated. That formula is two times pi times radius (r) squared plus two times pi times radius (r) times height (h) (the second formula below).

Now, we're going to assume that volume is constant. By using a constant volume, we can figure out what gives us the most volume for a given surface area. If it sounds backwards, don't worry. It'll work out, I promise. Now we can begin work.

Our first problem is that in the surface area formula, we have two variables, r and h. Since we said volume was constant, we can solve the volume formula for h. We find that h = V/pi r squared. We can then plug h into the area equation. So far, we've only used algebra to get our new formula for area.

Now for the calculus. We want to take the derivative of A (area) with respect to r (radius). If that sounds messy, here's what you need to know. We want to know how the area changes if the radius changes. You can imagine that if radius increases, area increases. The derivative just gives us a formula that tells us exactly how the radius affects the surface area. We write the derivative as dA/dr (area to radius). We usually use derivatives to find minimum and maximum values of functions. In our case, we want to find when the surface area is at a minimum. This happens whenever the derivative is zero (don't worry about why this is if you don't know for now; you can always look it up later). On the first line, I've found the derivative; on the last line, I set it equal to zero.

We can now solve for V in terms of r.

This means that when surface area is minimized, volume equals two times pi times radius cubed.

Using this and the original equation for volume, we can find out which value of h gives us the smallest surface area for a given volume.

This means that when the can is as wide (2r = diameter) as it is tall, we get the lowest surface area for a given volume. In other words, we get the most volume for a given surface area. Unfortunately, I haven't noticed a lot of cans this shape. Most of the cans I see are tall and skinny, even though that shape isn't optimized.

Let's try cereal boxes. This math will get a little heavier, but the idea is the same.

The volume of a box (any box) is given by length times width times height. The formula for surface area is a little more complicated. Just know that l is length, w is width and h is height, and the second formula below will give you surface area.

For volume, we have three variables, which is two too many. Again, let's let volume be constant and solve for l (this will eliminate one variable). We get that l = V/wh. Now we plug that into our area equation.

Again, we've done no calculus yet, but here it comes. We're going to take two different derivatives. Sounds messy, but I'll explain. First we want to find the derivative of surface area with respect to height (how does area change with height?) (dA/dh). Second, we want to find the derivative of surface area with respect to width (how does area change with width?) (dA/dw). You might notice the funny symbol I used for d. Don't worry too much; it just means I have more than one variable. Just like before, though, I want to set them equal to zero to find the minimum value for surface area.

 

If you know how to take derivatives, know that these are partial derivatives. And if you know how to take derivatives, then know in the first case, I assume w is constant, and in the second case, I assume h is constant.

When I set dA/dh equal to zero, I solve it and get that V = wh^2.

When I do the same for dA/dw, I get V = hw^2.

Since volume is constant, I can set them equal to each other. I find out that h = w, so I make them both equal to s, a new variable to represent either h or w.

With s, I get the two equations for volume and surface area below. Again, in surface I have two variables and want to get rid of l by solving my volume equation. This gives me l = V/s^2. If I plug this into the area equation, I get the result at the bottom.

I want to know how surface area changes with s, so I need to take the derivative (regular this time) to find dA/ds. Like always, I want to set that equal to 0.

When I work that out, I find that surface area is minimized for a given volume (or volume is maximized for a given surface area) when volume equals s-cubed, which means all three sides are the same length.

The shape that optimizes this problem, then, is a cube. Again, I don't see many cereal boxes this shape. Most I see are tall and skinny, just like the soup cans.

So what gives?

Which can did you think would hold more: tall/skinny or short/fat?

What about the box: cube or tall rectangle?

If you're the average person, you picked the tall one both times. It's a psychological thing that makes us think the taller container holds more; we completely ignore the other two dimensions!

Companies know this. That's why they don't bother optimizing their containers. They know they sell less in a box that costs the same to make while making you think it's more than it is.

So are soup and cereal companies evil? Not necessarily, but they do know how to play with your mind.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

I first learned optimization problems from my high school Calculus teacher Jan Haynes. 

The images here were produced by me using Noteshelf for iPad.