Surface optimization for a volume

Question

One of my children received this homework and we are a bit disoriented by the way the questions are asked (more than the calculation actually). This is exactly the wording and layout of the homework:

Consider an aluminum can: for example, a Coke can. Do such cans have the right dimensions? What does "right" mean? Why do other products come packed in containers of other shapes?

Question:

• An ordinary can has a volume of 355 cm3. What shape will hold this volume of liquid using the least amount of aluminum (minimising surface area)?

• Demonstrate your conclusion by comparing 2 different shapes in addition to a cylinder.

• Consider defining suitable variables and stating any and all assumptions you make.

• Use differentiation to find the value of your variable that minimizes the volume of metal used to make the container. Is there another method you can use to justify your model?

• Are real containers this shape? Why or why not?

• Discuss which model best fits the actual container, giving reasons for any differences, if they exist.

You are then informed that the circular top and bottom of your drink can has thichtess 0.2mm but that the curved surface has thiclvtess 0.Imm.

• Nist Inc. would like to launch a new sized can that has a capacity of 200 ml. Using your model, find the dimensions of the can that would minimize the volume of metal used to make the can.

• Do you think that Nist Inc. would use these dimensions? Why or why not?

First, keep in mind that they have only seen derivatives so far (no integrals yet). Second, please understand that we are not native English speakers so decoding the sentences is part of the problem.

Finding the optimal height and radius of the cylinder is quite trivial:

• Find the constraint equation (volume in terms of height)
• Find the optimizing equation (surface Area).
• Plug the height of the volume equation into the optimizing equation
• Derive that equation and make it equal to zero
• Solve it for the radius and plug the radius found in the height equation.
• The answer will provide you the ideal height and radius to optimize the surface area of the cylinder.

So far so good, it seems that the actual cans are more or less optimal if simplified as pure cylinders (but the homework seems to consider there is another "model" that could be used. I went and check on the internet with no success. This model seems the obvious one).

Now, does it have the "right dimensions"? "what does right mean"? and "why do other products come packed in containers of other shapes"? It seems to be anything but pure math questions and it seems to be dependent on how you see things (storage, drinking convenience,..). As I'm sure I'm wrong, I can't possibly understand what answers are expected here. Maybe I'm missing the point to make it all clear, at once.

The "right" dimensions? For a soda can, it's pretty close to the ideal measures yes but is it the expected answer? No clue.

"What does right mean"? Well... I don't know what this question is really asking. In terms of Maths? In terms of storage issues? In terms of practicality? In terms of costs?

"why do other products come packed in containers of other shapes" Same here? Cubes are easier to store I guess? Flat tops and bottoms allow a more convenient way to stack things? But I'm pretty sure I just don't get it.

Moreover, this is an intro text and a series of questions are coming just after. I just don't know if these questions in the intro text are rhetorical or if they are already meant to be answered. Which would be strange as there is no constraining data yet.

Now come the actual questions:

"What shape will hold this volume of liquid using the least amount of aluminum (minimising surface area)?" That would be a sphere, with no doubt. But how to prove it in a trivial and absolute way? It's seems intense.

"Demonstrate your conclusion by comparing 2 different shapes in addition to a cylinder". How a non comprehensive comparison would demonstrate anything? Here too, I don't get it. I could make the calculation for a cube and then a cylinder and then a sphere to show that it would decrease the surface area for a given volume but that wouldn't be a proof of anything, would it? All I would be able to say is that "it seems" that the more we go towards a shape with an infinite amount of sides (perfect curvature), the more we will optimize the surface area. But that doesn't demonstrate anything, especially if we just take 2 other shapes to get to that conclusion.

"Is there another method you can use to justify your model?" Besides using the derivative to find the optimal height and radius? I can't seem to find another. Are we talking about the sphere model or the cylinder model now?

"Are real containers this shape? Why or why not?" Sphere or cylinder? What shape are we talking about now?

"Discuss which model best fits the actual container, giving reasons for any differences, if they exist." Which model is there that is obvious, besides the derivative? It feels like the questions are not specific enough.

"Do you think that Nist Inc. would use these dimensions? Why or why not?" Here too, I'm lost. I can calculate the dimensions just fine but it seems that I have to go through the same reasoning than all the questions above...which are already confusing.

My question is somewhere between a math question and an understanding question. But if I go and ask on another forum dedicated to English, they might be unhelpful because of the math aspect of that homework.

So, all in all, it seems to be the best place to ask the question. I'm sure I just miss the point of that homework which makes all the questions quite obscure to me. Maybe some hint will help deblocking the situation at once. I feel like I'm missing something obvious. That's what I'm asking for.

I'm probably sure this post will make some people laugh and make a clown out of me but be assured that the language barrier doesn't help.

Thanks in advance.

Answer 1

In business "right" means (Profitability of your answer - Profitability of obvious solution) / Time x hourly salary is large. Again this formula is right business-wise.

To sum up an interesting article discussing all your questions as well as the optimization problem. A cylinder is a compromise between:

• surface volume ratio (cost of the material)
• shape easy to manufacture (to build a cylinder you wrap up a rectangle and add 2 disks)
• flat top and bottom for stacking up the products
• rounded edges to minimize the stress and therefore minimize the thickness of the sides (material used)

Why does the sphere minimize the surface / volume ratio? Well the rigorous proof is 'PhD-complicated' I think. It is intuitive for the same reasons than a circle minimize the perimeter / surface ratio in dimension 2.

Answer 2

Since you're asking for an opinion, here's mine.

I too find this list of homework questions somewhat disconcerting. I think I understand the point of the assignment: embed the routine calculus problem in a "real life" situation.

That said, I think the parts of the question intended to be "real" are themselves both quite artificial and much too vague.

If the course really wants to teach about building mathematical models for real world applications then lots of time should be spent discussing what makes a good model, with correspondingly less time to devote to the routine calculus "word problem". If that were actually explicitly part of the curriculum and had been covered and discussed then something like this (with more tightly worded questions) might make a reasonable exercise.

No one sells soft drinks in spherical containers. It might be interesting to ask how much extra material you'd use for a hexagonal prism the same height as the coke can - maybe you don't want the can to roll. And you could pack those cans into a box with less wasted space (how much?). That would be artificial too, of course, but maybe interesting.

As you clearly indicate, asking about the "right model" is nonsense without much more information - you have to specify some reasonable optimization criteria - more than minimizing material cost. Maybe hexagonal cans cost more to manufacture.

If my son or daughter brought this home I'd use the opportunity to discuss what the instructor was intending to get at, criticizing how s/he constructed the questions, then constructing the answers the instructor was hoping for.

Answer 3

Why a cylinder? why not a cube or a sphere or a parallelepieped?

A sphere has the best surface to volume ratio, but then where would you put the opening, and they would roll of the shelves, etc.

If it is a cylinder, your variables are height and radius. If it is not a cylinder, then your variables might be something else.

The optimization problem

$v = \pi r^2 h\\ SA = 2\pi rh + 2\pi r^2\\ A = 2\pi rh(0.1) + 2\pi r^2(0.2)$

Using volume as the constraint optimize $A$

You will come up with some measure of $r, h.$

Now measure and actual can, does a standard can approximately meet these dimensions. If not, might there be another reason.

If you change the volume from $335$ ml to $200$ ml how does that change $r,h.$ I will suggest that the optimized cylinder for the smaller volume should be similar to the larger cylinder.

When you see beverages in smaller cans in stores, are the similar to the larger can? If not, why do you think that might be?