I am giving an exam in my calculus class today. My students are worried about antiderivatives and definite integrals and the fundamental theorem of calculus, but I am not. I, instead, am worried about being a presence in the classroom so that they will fight the urge to look over at the paper next to theirs. While it is important to ensure academic honesty, it is also extremely boring.

As I roam around the classroom, my mind wanders. Inevitably, I think about all the grading I’ll have to do after the exam. Therefore, I search for anything else to think about. At this point, I notice all the water bottles and coffee cups on my students’ desks. They all seem to be approximately cylindrical in shape and have similar dimension; however, they are not exactly the same.

Upon noticing this, I wonder why they aren’t the same. Should they be? What would be the optimal dimensions of a bottle or cup? So, let’s find out.

**What is the purpose of a bottle?**

In order to determine optimal dimensions, we should first find what we want out of a bottle. Well it should hold liquid. In fact, it should hold a specific amount of liquid. Instead of working with the general bottle, let’s work with a 1 liter (1 quart) bottle of water. This should hold exactly 1 liter of water. Other than that, though, we don’t have a lot to go on.

In general, we could hold a liter of water in all types of different shaped containers in all types of different dimensions. That really doesn’t limit things, so let’s think about what we would want. These are the things I came up with:

- The container should stand without falling over if you set it down, so it should have a flat surface.
- It should fit in your hand, so the sides should be roughly circular. There may also be a limit on how thick it can be in order to fit the average person’s hand.
- The top should have a cover or cap in order to help prevent spilling. While the cap doesn’t need to be flat, we will assume it is in order to have a shape to work with.

In particular, the general shape of a water bottle should be cylindrical. As I look around at the bottles on my students’ desks, this is indeed the rough shape of all of the containers I see. Therefore, we will continue under the assumption that we have a cylindrical container.

**Goal in design**

While we know that we should make a cylindrical bottle, we can still have many varying heights and radii that give a volume of 1 liter. In order to think about an optimal shape, we must also have some goal in mind. My thought would be that we should use the minimum amount of material possible. The reasons for this, are

- This would result in the least amount of waste or litter resulting from each water bottle. Since, plastic waste is a major problem, this seemed like a worthwhile goal.
- This would also result in the lowest material costs for manufacturing the item.

Since at least one of these seemed worthwhile regardless of your outlook, we will try to find the shape that will result in the least amount of material being used.

**Modeling**

Now that we’ve determined the restrictions and goals in making a water bottle, we will construct a mathematical model. In order to do so, we draw a cylinder below.

Here we let \(r\) be the radius and \(h\) be the height of the cylinder. In this case, we must have that the volume is 1 liter. Since we will be finding the distances of radius and height, we will convert this to 1000cc (cubic centimeters, 61in\(^{3}\)). Since the volume of a cylinder is given by \[V=\pi (r)^{2}h,\] we have that \[\begin{align*} 1000&=\pi (r)^{2}h \\ h&=\frac{1000}{\pi (r)^{2}}.\end{align*}\]

If the bottle has a uniform thickness, then the amount of material used will be proportional to the surface area of the bottle. Therefore, if we minimize the surface area, we will minimize the material used. Since the surface area of a cylinder is given by \[S=2 \pi r h +2 \pi (r)^{2},\] we will need to minimize this over an appropriate domain.

We now need to find the domain that makes sense. Since the radius and height are distances, we will say that \(r \geq 0\) and \(h \geq 0\). Furthermore, if \(h=0\) or \(r=0\) we get \(V=0\), we will also assume that \(r \neq 0\) and \(h \neq 0\).

We also want to limit the radius in order to ensure that the average person could hold the bottle. Here we will work with an average human hand length of approximately 18cm (7in). In order to grip a cylinder, we would curve our hand in a circular manner. However, we don’t need to wrap our hands around the entire bottle to get a good grip, so we will assume we have to be able to grab approximately half the circumference. This would mean a radius of 5.7cm. We will assume a maximum radius of 5.5cm (2.17in) in order to make the grip a little more comfortable.

We now have that the appropriate domain for the radius is \(0 < r \leq 5.5\). Furthermore, if we substitute for \(h\) in terms of \(r\), we are left with the problem: Minimize \[S=\frac{2000}{r}+2 \pi (r)^{2},\] subject to \(0 < r \leq 5.5\).

Now, recalling work from calculus, we would have that the minimum, if it exists, would have to occur at a critical point. In order to find these, we take the derivative of surface area with respect to \(r\) and get\[ \frac{dS}{dr}=-\frac{2000}{r^{2}}+4\pi r.\] The critical points are where this is zero or undefined, so we find \(\frac{dS}{dt}=0\) when \(r=\sqrt[3]{\frac{2000}{4\pi}} \approx 5.4.\). Now, since the function is only defined on the interval \(0 < r \leq 5.5\), we would have that the function ends at \(r=5.5\). Therefore, the secant lines from the right at this point do not exist, so that tangent does not exist. Hence the derivative is undefined at \(5.5\).

In order to determine which if any of these will give us a minimum surface area, we look at the sign chart of the derivative. In doing so, we have that \(\frac{dS}{dr} < 0\) on \(0 < r < 5.4\) and \(\frac{dS}{dr} > 0\) on \(5.4 < r < 5.5\). This tells us that the function decreases until \(r=5.4 \) and increases afterwards. Therefore, the minimum surface area will occur when \(r=5.4\)cm (2.12in)

### Interpreting our results

Through solving our model, we have seen that we will use the minimum amount of plastic for our 1 liter water bottle if we let the radius of the cylinder be 5.4cm. We should note that this is very close to the upper limit we placed on the radius of the bottle. Therefore, if the bottle was meant to hold much more water, we would have to stop at this maximum radius in order to use the minimum amount of material.

Now, what would this bottle look like? Well, if the radius is 5.4cm, that means the diameter of the bottom and top is 10.8cm. This would then result in a bottle with a height or 10.8cm. That is, the thickness of the cylinder should be the same as the height. If we compare this to an actual 1 liter bottle, we get something like this,

Note that the actual bottle is thinner than the optimal water bottle. Therefore, if we were to change the dimensions of a water bottle, we could indeed reduce waste while still having the bottle satisfy our requirements for use.

**Conclusion**

By looking at the purpose of a 1 liter water bottle, and identifying a goal of minimizing material used in construction, we were able to find the dimensions of the optimal water bottle. However, when comparing our results to actual bottles, we noticed that they did not match. In fact they were significantly off.

This observation leads to the question, why is this the case? Is my analysis not correct? If this is the case, is it because my assumptions were incorrect? If so, which assumption? While I don’t have definitive answer for you, the likely culprit is that bottles are not made in order to minimize material used.

I do not work for a bottling company, but I would venture to guess that thin tall bottles appear to hold more liquid than short thick bottles. That is, psychologically, by using a different shape the consumer perceives more value from a tall thin bottle, hence, companies are able to charge more for the same amount of water. If you happen to be involved in bottle design, please comment below and let me know why companies chose the current shape. I’d love know why companies continue to go with a non-optimally designed bottle.

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