Optimizing Cups to Get Every Ounce of Sauce Possible

[@keaganhurter2550]

i love your videos, they're exactly the type of math problems i think of. and it's just so great to see my questions answered

[@MathTheWorld]

Thank you. If you think of problems like the ones we think about, I would love to hear what other problems you have thought about. Maybe some of them would make good videos.

[@Fire_Axus]

your feelings are irrational

[@HawkulusQuest]

I was one of the people who commented that you were not the only person who cares about this problem. I spent quite a bit of time coding this problem in python, and I even tested a few different height to radius ratios, but with this video you have gone above and beyond. And I am so here for it! Thank you, I can't wait for part 3!

[@MathTheWorld]

Amazing!

[@mujtabaalam5907]

There's no reason for the cup to be limited to a frustum-type shape; you can use variational calculus to find the optimal curve with the constraint of surface area.

[@MathTheWorld]

Yes exactly! That will be part 3!

[@tttITA10]

For people who were not too happy about the explanation as to why maximizing the cross sectional area isn't the same problem here, by the Theorems of Pappus-Gulding, the revolution volume IS actually proportional to the cross-sectional area, the problem being it is also proportional to the average distance of the points of the object (or, in this case, one half of it) to your axis of rotation, so, since changing the shape of the object (as if to maximize the area) may also change the average distance of the object's points to the axis, the problem is more subtle then just maxizing the cross-sectional area. His example got confusing because he changed the axis of rotation (instead of showing the equivalent change in shape), while, in the paper cup problem, the axis of rotation is determined.

[@FunctionallyLiteratePerson]

Thank you for this! I was thinking along the same line but couldn't remember exactly the reason why

[@MathTheWorld]

Thanks for bringing in Pappus. Making that connection it's a good way to understand why maximizing the 2D area is not the same as maximizing the volume.

[@kamo7293]

I just realised a 2 parter on sauce cup optimizing is also a good intro to derivatives and now partial derivatives. seriously when you said "multi variable calculus requires partial derivatives" I was kinda mind blown. partials were taught to me without context, reason, and use case. it's just something you need to know and that's it. and that never sit right with me. it made understanding them in the first place for me difficult.

[@MathTheWorld]

That was my experience with a lot of the math I learned (secondary and university). I really try to motivate everything I teach, and as much as I can, motivate it by the power to makes sense of a real-world situation. That is one reason why I made this channel, to show people how the math that they learn in school or college is actually powerful and useful. Maybe optimizing sauce in an Arby's cup is not the most motivating context, I certainly have others for derivatives and partial derivatives. I think it's enlightening though that something as simple as an Arby's cup has, has such mathematics hidden behind it.

[@brodymiller9299]

One thing to point out is that the term "Viscosity" is not accurate for how high a sauce can extend beyond the rim. In this case, if the sauce is a Newtonian fluid, the fluid will always overflow the edges (though it may take a long time, such as the material "Pitch"). Ketchup is not a Newtonian fluid, but rather a Bingham Plastic. Bingham plastics do not flow when under a certain shear stress, known as the "Critical Stress" or "Yield Stress". While it can be correlated between similar fluids with viscosity, it is not the same, and a Bingham plastic can have a low viscosity. As an example, Coal slurry has a lower viscosity than printing ink, but a higher critical stress. For ketchup, I have seen critical stresses in the range of 15-30 Pa, which would give a support angle of 35 - 45 degrees at a 1.5cm radius. I'll try and make a longer write up once I do some additional math.

[@chain3519]

I haven't gotten to that part of the video yet, but wouldn't viscosity not even apply here? I thought viscosity whether that be Newtonian or otherwise was a mapping of fluid velocity gradient across a surface. In this case the sauce should have zero velocity and therefore the gradient velocity of the ketchup should be zero everywhere. The mechanism that allows sauce to exceed its volume I believe is what you are describing which I think means the sauce continuum is able to generate some internal shear stress along the skin to resist body forces

[@brodymiller9299]

@@chain3519 Yea, I did the math and it ends up being a sqrt curve from the outside in. I got a total volume of 97.4022cm^3 after all was said and done, with a bottom radius of 2.22cm top radius of 3.802cm, and an angle of 62.8 degrees. I used a 21.8 Pa critical stress since I found that in a google search.

[@bengoodwin2141]

I'm no expert here, but isn't surface tension involved here? Or is that a different phenomenon

[@brodymiller9299]

@@bengoodwin2141 Surface tension would have an effect, but it would be less than for water due to the other components in it. A quick google search found that water has a surface tension of ~ 72 mN/m, whereas ketchup is 30-40 mN/m.

[@Petch85]

If you liked this video Numberphile also just made a video on cones. "Cones are MESSED UP". There is a little less math in the video. But the video and result is fun.

[@tesseract2144]

I think it's possible to parameterize the angle of the sauce and find the optimal radius+ cup angle for every sauce viscosity possible

[@caked3953]

and he want's to adress the statics of the cup too. Slowly but surely we uncover the universal "paper cup with foldings out of a round pice" formula!

[@xavimourelo]

Another problem that might be worth explore is to see how toilet paper decreases in size, because when you do a 360º turn at the beginning the volume decreases very little, but as you approach the end each new turn reduces dramatically the amount of paper left, not being able to properly guess when is it gonna finish

[@MathTheWorld]

We have this one in the works!

[@ArthurKhazbs]

If only toilet paper manufacturers numbered the sheets!

[@friguspersona]

I absolutely love your content! Can't express it in words enough. Its great to see what I've learned applied to things I find weird (in a good way).

[@MathTheWorld]

Thank you so much. We really appreciate the positive feedback.

[@friguspersona]

: )

[@wavebreakerr142]

I optimized my sauce cups by blowing into the top and inflating the cup like a balloon. If I keep the rim intact, I'll end up with a highly optimized shape. Unfolding the rim expands the shape slightly, but the math starts to look like differential geometry and dynamical systems, which I'm not ready to mess with yet

[@MathTheWorld]

A couple other commenters have shared that strategy. It's one I hadn't thought about. I toyed with the idea of analyzing that strategy in part 3 that's coming up, but the math is just as difficult as optimizing the bowl shape when you open up the rim.

[@cadenbappy7408]

You should also make a video about the fastest a person could theoretically run.

[@Petch85]

Good work.If it is worth doing, it is worth overdoing. 😂

[@MathTheWorld]

I like that quote!

[@Starwort]

I'm pretty sure the cross-sectional area is proportional to the volume of the solid of revolution iff you square the magnitude of the coordinates on the cross axis (e.g., the x axis when revolving around x = 0) before calculating the area, but that's a non-linear transformation so might be harder to optimise for than just working through the problem in 3d (e.g. a straight-line side would be transformed into a quadratic curve for area determination)

[@cuwej8338]

As an european who has never been to arbies, I’m glad that these videos exist. Definitely need more of Math Overkill

[@MathTheWorld]

Thank you! Just wait, we have another one coming soon on toilet paper!

[@EdbertWeisly]

Next: instead of Math Overkill, its Math Slaughterhouse

[@dakerbal]

Excellent video. Something I did find a little unsatisfying in the previous video is that you went straight into the numbers/graphing instead of trying an analytical approach. Here you end up (unfortunately) with a cubic, which isn't very nice, but it does show (albeit this is already pretty obvious) that the optimum angle θ only depends on the ratio of the base radius to the slant length (r/l = a): sin³ θ + 2a sin² θ + (a² − 2/3) sin θ - a = 0 Sometimes an analytical approach gives better insights into how the solution would change as you change the parameters (although here it may not be as useful). Of course graphing is still super nice as a visualization.

[@MathTheWorld]

That is really nice. Yes, I should have emphasized that more. But on the bright side, you can use the general form you've created to solve the challenge problem at the end of the video!

[@blocksy6772]

Awesome idea for a channel, you definitely deserve to get more views! Wondering if you're thinking about adding some dynamic/recursive models as well? I remember we used to have a course in my economics program that started out with a silly question like 'What's the best way to eat a cake', given that you can either eat it now or save some for later, and it ended up covering some interesting models with regards to what the best balance was for consumption/saving, given that any consumption today makes you happier right now, but saving allows you to consume more at a later time. So you get really neat models that take time into account and get dependent on past actions/future considerations.

[@MathTheWorld]

Thank you so much for the positive feedback! I am sure we will end up with some dynamic models at some point. But I hadn't thought about modeling cake eating. If you have more information on that, send it my way. I've read a lot of economics, but mainly conceptual, not mathematical topics. This is one reason why I feel like I need an undergraduate degree in every technical field because it would open my eyes to so many ways that mathematics is used.

[@blocksy6772]

​@@MathTheWorld Sure thing, I looked around a bit and found a pdf file that explains the setup and shows you can work these models out in Excel, but there's a whole bunch around that explain it in greater detail if you look up 'cake-eating problem economics': ani.stat.fsu.edu/~jfrade/HOMEWORKS/Fin_Econ2/session05.pdf Obviously, the cake-eating problem is an analogy for a simple consumption/savings problem, and you can start adding all kinds of additional considerations into your models if you want. Hence, a lot of assumptions about 'utility' gained from consumption is very general, but you're free to use more or less whatever utility functions you want given the scenario you're trying to model. Also found a youtube video of someone who modelled this in excel and added uncertainty by using random time-preferences for the cake-eater, so you get different consumption behaviors depending on how much someone would discount their future consumption. He doesn't really explain it in detail but its pretty neat: kzbin.info/www/bejne/hqKZn5SCmbxkpac&ab_channel=EconJohn

[@alex.g7317]

When you love ketchup but too cheap to get an extra paper cup:

[@jorgefadul9110]

There are a few things to keep in mind with having that shape. The weight of the center of mass of the volume that is directly above the angled wall will exert pressure on the angled wall, therefore this shape could not be stable over time. And the rheology of the sauce could not be steady with that 30° angle

[@jcorey333]

This is fun to watch! I definitely wish I really understood multivariable calculus, but it wasn't too bad seeing the graphical solution to this problem. Of course, now I'm feeling hungry for some Arby's.

[@MathTheWorld]

Maybe you could audit my class. (If I ever teach it.) Although you know my office neighbor is better than me.

[@FunctionallyLiteratePerson]

It should be mentioned for the multivariate version that it is possible that the partials are both 0 are a local extrema and not global. You also want to check the edges of the domain, as partials might not equal 0 there due to domain restrictions but you still might have a greater value. One might argue you can ignore this if you know roughly where the maximum is and the edges of the domain are definitely nowhere near, but that only works in certain cases and not for everything.

[@MathTheWorld]

The once you bring up our valid. I debated how much to get into these, but since we had a good visual of the situation once we restricted the domain to the context we were working with, I thought it might be a bit much to emphasize those things ( or saddle points) since we could see from the surface what we were dealing with. I certainly could have hurt viewers thinking that finding the maximum on a 3D surface was always this easy.

[@VeteranVandal]

When you said "i thought I was the only person" i was like, "no shot, mate".

[@MathTheWorld]

Yep, that was pretty arrogant of me to think that no one else had thought of that.

[@bengoodwin2141]

I think it might be relevant to consider that the sides of the cup expand by unfolding, so they might act differently than just being a rotating curved... plane?

[@shy_dodecahedron]

Are you going to account for the different positions of fingers? Also will we account for the way he is transporting the sauce? I think moving sauce from side to side can help some sauce remain permanently in motion, never falling.

[@ecMathGeek]

If you vibrate the sauce fast enough, can you counter the viscous flow of the liquid? Seems like we need the science-guys of KZbin to investigate this absurd proposition.

[@ecMathGeek]

Alternative solution: send the sauce into freefall and there won't be a technical limit to how much sauce the container can "hold."

[@MathCuriousity]

Why did you change the axis of rotation?! By the way: your channel is AMAZING!!! Absolutely hats off to your level of both creativity and commitment to service and educating self learners!

[@MathTheWorld]

Thank you so much! Is your question referring to our argument about my optimizing the 2D area is not the same as optimizing the volume of revolution? You end up with the same shapes by rotating the axis or by rotating the area, but by rotating the axis it's a lot easier to see that one has more volume than the other because the small one is almost completely enclosed in the large one. Doing it the other way is just a little harder to see.

[@PassFirstPost]

I hope the next video includes surface tension

[@scott2427]

I wonder if you could ever do anything with signal analysis it would make for a cool video in my eyes

[@MathTheWorld]

Can you tell me more about what you are thinking about signal analysis? Do you have particular situations in mind?

[@scott2427]

@@MathTheWorld I had an idea about optmizing an audio signal algorithm to mimmic a human voice or how it can be used to correct data

[@VeteranVandal]

More like math oversauce.

[@raptokvortex]

For those interested, the optimum shape of bowl like containers for liquids is a truncated hemisphere.

[@MathTheWorld]

There might be a situation where that is optimal, but if you're trying to make a bowl with the minimal amount of material that holds the most volume, it isn't quite a truncated hemisphere. We're going to get as close as we can to that in our next video.

[@Petch85]

Link to the video with the ketchup testing?

[@MathTheWorld]

Here you go! kzbin.info/www/bejne/nWXFf5uOjZaiiNk Thank you for the reminder. I put it in the description now, like we promised.

[@rrrrmrmr]

But would it actually be possible to fold the cup to have that radius without cutting it?

[@MathTheWorld]

Yes, you don't have to cut it to get a different sized base. The folds are approximately triangles with the vertex starting on the edge of the base. So you can fold the triangles and have that vertex any distance away from the center. Well, up until physical limitations of folding the paper into an extremely small fold.

[@JuegosLasTazas]

Now remember the uneven ketchup and the ketchup that is put on top of another

[@siddharth_desai]

If you're allowed to change where the flat region of the cup ends, then surely, you're not limited to a flat region and a angled region. The angle of the walls could vary continuously to make a rounded bowl shape that could hold even more sauce. Anyone up for some calculus of variations?

[@MathTheWorld]

Yes that is the plan for part 3!

[@thebeardman7533]

The next problem sounds like a physics problem

[@ExzaktVid]

This is how our teachers expect us to use math

[@AZPStuff]

No desmos 3D 😢

[@MathTheWorld]

unfortunately no, but we use Geogebra 3D!

[@kelpR]

nice

[@MathTheWorld]

Thanks!

[@aggking7034]

great video! (first)