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.
Best Answer
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:
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.