Optimizing Cups to Get Every Ounce of Sauce Possible

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We heard the people. More Sauce. More Math. More Optimizing. In this Math OverKill part 2 video we tackle more saucy questions to optimize the amount of sauce in a condiment cup all while diving into multivariable calculus and partial derivatives. We address misconceptions about optimizing in 2D and tackle questions like "what is the optimal angle if we can overfill the cup?" and "what if we could change both the angle and the radius of the cup?" So join us even further down this rabbit hole in this week's episode of Math The World.
OverKill part 1: • The Optimization Probl...
Here is the link to the video made by RU-vidr ‪@CooperGiles-mv3jn‬ who solved the problem assuming the cone shaped overfill: • The Mathematically Opt...
Math The World is dedicated to bringing real world math problems into the classroom and answering the age old question “when will I ever use this?”
We use unique topics for algebra, trigonometry, calculus, and much more and go beyond context problems and use a technique called mathematical modeling to find solutions to real world questions and real world problems. These videos are great for students who plan to enter technical fields that require real world problem solving, and can be a great resource for teachers looking for ways to bring real world contexts into their classroom.
Instagram: / maththeworld
Twitter/X: / math_the_world
Facebook: / maththeworldproject
Email: MathTheWorld@byu.edu
Created by Doug Corey
Script: Doug Corey and Jennifer Canizales
Audio: Doug Corey
Animation: Jennifer Canizales
Music: Coma Media
© 2023 BYU

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28 май 2024

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Комментарии : 74

@keaganhurter2550 27 дней назад

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 27 дней назад

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 23 дня назад

your feelings are irrational

@HawkulusQuest 27 дней назад

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 26 дней назад

Amazing!

@mujtabaalam5907 26 дней назад

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 26 дней назад

Yes exactly! That will be part 3!

@tttITA10 26 дней назад

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 26 дней назад

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

@MathTheWorld 25 дней назад

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.

@brodymiller9299 27 дней назад

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 27 дней назад

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 27 дней назад

@@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 24 дня назад

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

@brodymiller9299 24 дня назад

@@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.

@tesseract2144 27 дней назад

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 25 дней назад

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!

@Petch85 27 дней назад

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.

@Petch85 27 дней назад

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

@MathTheWorld 27 дней назад

I like that quote!

@kamo7293 20 дней назад

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 20 дней назад

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.

@cadenbappy7408 27 дней назад

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

@alex.g7317 27 дней назад

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

@xavimourelo 27 дней назад

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 26 дней назад

We have this one in the works!

@ArthurKhazbs 18 дней назад

If only toilet paper manufacturers numbered the sheets!

@wavebreakerr142 25 дней назад

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 24 дня назад

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.

@EdbertWeisly 27 дней назад

Next: instead of Math Overkill, its Math Slaughterhouse

@friguspersona 27 дней назад

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 27 дней назад

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

@friguspersona 26 дней назад

: )

@Starwort 27 дней назад

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 26 дней назад

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

@MathTheWorld 26 дней назад

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

@dakerbal 27 дней назад

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 26 дней назад

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!

@VeteranVandal 27 дней назад

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

@MathTheWorld 24 дня назад

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

@blocksy6772 24 дня назад

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 24 дня назад

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 24 дня назад

@@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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Pq6naLdW1oE.html&ab_channel=EconJohn

@jcorey333 27 дней назад

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 25 дней назад

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

@bengoodwin2141 24 дня назад

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?

@PassFirstPost 19 дней назад

I hope the next video includes surface tension

@MathCuriousity 24 дня назад

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 24 дня назад

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.

@FunctionallyLiteratePerson 26 дней назад

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 25 дней назад

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 27 дней назад

More like math oversauce.

@raptokvortex 26 дней назад

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

@MathTheWorld 25 дней назад

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.

@johnacetable7201 27 дней назад

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 26 дней назад

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

@ecMathGeek 26 дней назад

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

@ExzaktVid 27 дней назад

This is how our teachers expect us to use math

@scott2427 25 дней назад

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

@MathTheWorld 25 дней назад

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

@scott2427 24 дня назад

@@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

@thebeardman7533 27 дней назад

The next problem sounds like a physics problem

@JuegosLasTazas 27 дней назад

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

@siddharth_desai 26 дней назад

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 26 дней назад

Yes that is the plan for part 3!

@rrrrmrmr 27 дней назад

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

@MathTheWorld 26 дней назад

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.

@AZPStuff 27 дней назад

No desmos 3D 😢

@MathTheWorld 26 дней назад

unfortunately no, but we use Geogebra 3D!

@Petch85 27 дней назад

Link to the video with the ketchup testing?

@MathTheWorld 27 дней назад

Here you go! ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-g4bNhXX1oRw.html Thank you for the reminder. I put it in the description now, like we promised.