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The Optimization Problem No One Cares About But My Son

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Do you love sauce as much as my son? Then you will love this video. Here we tackle a calculus optimization problem to find the best angle to unfold those little paper condiment cups so you can maximize the amount of sauce it holds. We do this using some trig and algebra to begin with and also take a calculus approach with volumes of revolution and integration.
Sauce Overkill Part 2: • Optimizing Cups to Get...
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.
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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

Опубликовано:

20 авг 2024

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Комментарии : 1 тыс.

@MathTheWorld 3 месяца назад

I want to thank all of you for your comments and questions. I love the extensions that people have suggested and ideas others have offered. I (or we) haven't had time to respond to all, but here is the follow up video! Math Overkill Part 2: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jSR-QR2GRqQ.html

@itsyo42 3 месяца назад

How many of the creases do I need to unfold for that angle?

@NathanK97 3 месяца назад

that all is a great theoretical answer, but practically how many folds do you have to untuck to get closest to the optimal angle?

@Quxi 3 месяца назад

Hi, I don’t know if somebody asked this, but I feel like you’re overcomplicating the problem. In my mind if we’d integrate in a polar coordinate system from 0 to 2π we could simplify the problem to optimizing the area of our slice, which simplifies the problem. We can see the area of our slice being the area of a rectangle plus the area of the triangle, which we can describe as height * radius + 0.5 height ^ 2 * tan θ from here my math can be rusty, but if we do a derivative by height we get 0 = radius + height * tan θ, and a derivative by θ we get 0 = height * radius + 0.5 height ^2 * (sec θ)^2 which we can substitute to get 0 = 0.5 * height ^2 * (sec θ)^2 - height ^2 * tan θ, which assuming height is positive leaves us with 0 = 0.5 (sec θ)^2 - tan θ, which, if I didn’t make a mistake means that the optimal angle is π/4 regardless of size. The follow-up question of a sauce pile is harder because of the physics of what kind of a cone can the sauce form, but still we can simplify it to 2D slices of instead of integrating disks along y we integrate slice rotating around y.

@BirnieMac1 3 месяца назад

We need more information to do the homework, it’s a wetting angle problem so don’t we need to know the surface tension of the sauce? I want those stickers dammit, we’re doing it properly

@FinleyRichard 3 месяца назад

Wait 8 kids, you are the math problem guy

@popemon7608 2 месяца назад

He must have calculated it's Cheaper By the Dozen

@HawkulusQuest 3 месяца назад

Your son is not the only one who cares! I wrote a python script to solve this very problem in 2021! I found it and dusted it off, and after plugging in your dimensions and accounting for the fact that I used the angle of the walls with the table as theta, I got the same answer! I used the calculus method by summing up tiny disks numerically without writing out the actual equation. Good to know someone else couldn't sleep without knowing this, and thanks for sharing it with the world! P.S. the question actually arose when my wife showed me a "life hack" that those paper cups were actually designed to do that. After some patent research I discovered that that is not true. It's just a very economical method to form paper cups from flat paper. But it did prompt me to ask what the optimal angle would be.

@HawkulusQuest 3 месяца назад

Also, for those who care, the total sauce capacity increases by 88% in this example, assuming I still understand the code I wrote 3 years ago. Not quite double, so I'm not sure I'd bother compromising the structural integrity of my cup, especially since you must also accurately gauge the angle to reach that 88%.

@dillonstrange5152 3 месяца назад

XD I think its great that you took the time to do "some patent research". Not, I think, to say "you are wrong and I'll prove it" but just to find out because you can.

@Nano-n 3 месяца назад

may I have the codes if it still available? I'm still new to python and im just figuring things out.

@HawkulusQuest 3 месяца назад

@@Nano-n Sure thing! I'll add some better commentary and send it when I get home today.

@Nano-n 3 месяца назад

@@HawkulusQuest thank you for the reply

@carlospena98 3 месяца назад

8 KIDS???? IN THIS ECONOMY?

@itsyo42 3 месяца назад

Free sauce is free sauce

@404-Error-Not-Found 3 месяца назад

When you know math, you can afford it apparently

@MAYBEMAYNOTBE2 3 месяца назад

He knows math don't question him😅

@gideonk123 3 месяца назад

@@itsyo42which sauce are you referring?

@itsyo42 3 месяца назад

@@gideonk123 mayo

@rithishkr1697 3 месяца назад

This is the kind of question that should be asked on math exams to get students even a little bit less scared about the problem

@Kokurorokuko 3 месяца назад

I think it's quite common for olympiads to have these types of questions.

@magicmeatball4013 3 месяца назад

@@Kokurorokukoolympiads having calculus LMAOOOO you innocent soul

@carultch 3 месяца назад

Usually in an introductory calc class, they try to give you optimization problems that can be solved analytically to an exact solution, and fairly easily without tedious algebra beyond the scope of the class, instead of relying on numeric methods. It can be very hard to come up with examples that simplify nicely, and this example, unfortunately didn't.

@Kokurorokuko 3 месяца назад

@@magicmeatball4013 what?

@magicmeatball4013 3 месяца назад

@@Kokurorokuko the math Olympiad has 0 calculus on it, it’s innocent to think that scary calc problems would appear on olympiad level stuff

@yanniammari1491 3 месяца назад

8 kids? bro keeping the fertility rates UP on his own

@gogo8965 3 месяца назад

the optimal technique

@andersskorstengaard4255 3 месяца назад

Bro found the optimal impregnation time slot, with 8 nerd stickers to boot

@caspermadlener4191 3 месяца назад

Practising multiplication

@James2210 3 месяца назад

7 boys and 1 girl. Can you imagine? that's like a 0.1% chance

@caspermadlener4191 3 месяца назад

@@James2210 The chance of getting 0 or 1 girl out of 8 children, assuming a fifty-fifty chance of getting a boy or a girl, is (1+8)/2⁸≈4%.

@Funnymoney101 3 месяца назад

Situations like these are where I most enjoy using math, not in finding an answer to a question that I know has already has been solved but in finding an answer to a question no one has asked.

@omerelhagahmed551 3 месяца назад

In fact it is very likely asked and answered intensively or it's very hard on higher levels

@setusof 3 месяца назад

Why human wanna be unique ? 🤔

@seeker296 3 месяца назад

Bc otherwise we're too easy to replace @@setusof

@alex.g7317 3 месяца назад

@@seeker296you say that like it’s a guarantee

@badabing3391 3 месяца назад

@@alex.g7317its close to one

@abduking. 3 месяца назад

8 kids💀💀💀bros trying to make a math clan

@pimage8047 2 месяца назад

A math class.

@GregFRDT 2 месяца назад

Families in math problems be like

@flamingfire0 2 месяца назад

Bro IS the math problem

@ben_26 3 месяца назад

the funny thing is, when i was younger i used to go to a frozen yogurt place where you could sample frozen yogurt in cups like this, and my older brothers friends would sometimes try to get the best angle of the cup to get the biggest sample. so yes, this question has in fact been asked before

@siliconhawk9293 3 месяца назад

average day of a mathematician, trying to prove something for days because you couldn't find any relevant research on it. then once you are done you figure out some guy proved it 50 years ago but it was a different domain of problem that didn't coincide with your domain but since the domain is the same the proof is the same as well

@TheBlueArcher 3 месяца назад

@@siliconhawk9293 you should have tried to fit the word domain in a few more times in that comment 😂😂

@HagalazI 3 месяца назад

When I was younger I contemplated the optimal angle. I found that I could not establish an intuition about the problem and surrendered. Thanks to you and Wolfram for doing all the legwork I wasn't going to.

@MathTheWorld 3 месяца назад

You are welcome! Do you have any other problems that have stumped you?

@HagalazI 3 месяца назад

@@MathTheWorld Not any other that I can think of. I thought about this problem some more. I think something that would get me closer to getting an intuition would be to think of the problem in terms of two cylinders: The smaller one in the center, not accounting for the flare, and one which encompasses the whole cup. Maybe that means this problem could also be represented as: 1/2(BigCyl-SmallCyl) + SmallCyl Is there a reason this doesn't work?

@v0id_d3m0n Месяц назад

@@HagalazI I think you'd end up with the same / very similar calculations to solve it because you'd have to take into account the heights of the cylinders and the radii as you expand the cup, which would probably still involve using the angle theta.

@malvoliosf 3 месяца назад

Don’t forget, the amount of sauce you can pile above the rim gets bigger with the CUBE of the radius of the rim, up to a certain point...

@dougcorey3830 3 месяца назад

Great Point! I (the script writer) was just thinking about it leveling out, like water that is over the brim of a cup, but if it keeps a somewhat cone shape, you could get a lot more sauce and it would impact the optimal angle. I have thought about doing a video on the optimal angle of an iceccream cone (that is the actual shape of a cone) if the ice cream above the brim of the cone is a hemisphere. I think it would be much more like a plate than a typical ice cream cone.

@malvoliosf 3 месяца назад

@@dougcorey3830 With granular substances like sand, there is a thing called “angle of repose”: you can pile up sand as high as you like, and for a particular kind of sand, the heap will be a cone with a particular angle. With semi-solids like sauce, it’s different, I think because it’s about surface tension and viscosity fighting gravity...

@dougcorey3830 3 месяца назад

@@malvoliosf I know about the angle of repose, but I don't know anything about semi-solids (except how good they can taste!). Thank you for the lead. I will venture over to a colleagues office in fluid dynamics and maybe they can help me understand it.

@WaffleAbuser 3 месяца назад

@@malvoliosfWith ketchup being a non-Newtonian fluid (seriously) I think treating it as a collection of particles might still be applicable!

@bbgun061 3 месяца назад

I don't think think so. When it's sitting in the cup, it's not under any pressure. So it's non-newtonian properties won't have any effect. There's something similar to angle of repose for thick liquid suspensions called "slump." (It's used to measure concrete...)

@5eurosenelsuelo 3 месяца назад

Optimization problems are always super interesting. They are a great tool for teaching and learning. Regarding the question at 8:25, I'd propose that rather than fixing an arbitrary value such as 5 mm over the rim, the problem can instead take into account the angle of repose which I think is a little more realistic and it is also an excuse to teach other interesting concepts.

@MathTheWorld 3 месяца назад

Thank you we agree!

@bixenteartola6777 3 месяца назад

By considering an angle of repose of 45 degrees and that the shape above is a spherical shape (the top part of a sphere that is cut such that the angle at the base is the angle of repose), I found an optimal angle of about 44 degrees. So the approximation of 45 degrees is even more accurate! But I wonder if a dome shape is really accurate, perhaps is would be closer to a cone? Also the angle of repose is not exact. But anyway, adding a pile on top added only about 3 degrees to the optimal angle, so any shape or angle of repose will still give a value close to 45 degrees probably.

@CooperGiles-mv3jn 3 месяца назад

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-g4bNhXX1oRw.html - I made this video in which I did just that!

@bumpty9830 3 месяца назад

I spent years focusing on optimization as an R&D engineer in the aerospace industry, and the most interesting lesson I learned was that your initial assumptions about the constraints of the problem are almost always wrong. In a toy problem like this with literally one degree of freedom in the design space, it's easy enough to nail down. But when you run a real-world optimization problem, your first results won't be about an optimum design, but about design space assumptions that need to be adjusted.

@cappincrrunch4493 3 месяца назад

Dude there is literally no reason on earth you should listen to me but this is exactly what math education needs. Application. I didn't know how to turn a question about the world into a math problem until I was an adult. And I didn't appreciate what I could do with math until that point. And now I love figuring out an equation for a real life scenario even when I cant solve the darn thing. I think this content is going to help a lot of people grow in their education. Much love and stay awesome!

@MathTheWorld 3 месяца назад

Thank you so much! I think there is a reason to listen to you, because I believe this is what Math Education needs as well. I have some empirical evidence, since so many of my calculus students appreciate my efforts to show them how powerful the tools are that I am teaching them. I actually think we have a better chance of changing things by starting an extra class in school (especially high school) called Real-World Problem Solving or maybe Modeling and Problem Solving. It is so hard to change the math curriculum, but we could start by showing kids some of the power and coolness in another class that isn't hampered by the "teach for the test" syndrome.

@v0id_d3m0n Месяц назад

I think there's already a lot of this already (in the UK at least). But perhaps the teaching part is lacking -- allowing/pushing/teaching students to think about the right method to solve a problem. Often in lessons, you're already in "trig mode" or "linear algebra mode" etc so you skip the hard part of deciding what method to use. Then in the exam, you're left alone and confused. I think more practice in independent problem solving where the method isn't immediately obvious from context / explicitly provided would be great at helping people enjoy and learn maths.

@dbarcene 3 месяца назад

Mathematician: Let's calculate the optimal unfolding of this topology... Phycisist: If we assume the cup to be cilidrical cone... Engeneer: Fill more cups. Soucer go brrrrrr!!

@concerningindividual629 3 месяца назад

Mathematician when the Arby's ends up closing because they spent too long trying to find the optimal cup angle (time was an unaccounted variable)

@scragar 3 месяца назад

My niece had a similar problem. My solution was to grab a cup meant for drinks, rip the top of it off so it's shallow enough to dip stuff, and use that. Much faster and easier.

@siliconhawk9293 3 месяца назад

computer science guy - well since i am too lazy to do maths and stuff. let me make a simulation and figure out the best way to do it. 1 day later - fu*k where is f the bug in the code. why the f did it crash. huh there is a fire in my pc.

@bumpty9830 3 месяца назад

As an R&D engineer I worked closely with both physicists and mathematicians, and my experience was more like: Mathematician: I've written a theorem and emailed each of you a PostScript copy of the paper including the proof. It's a shame we're constrained to real-valued angles. Physicist: Okay, but we are constrained to real-valued angles, so here's the optimal result. Engineer: We built a prototype yesterday. If you want to get closer than this, you're going to need a new factory for precision construction.

@empty_thesaurus 3 месяца назад

Programmers: ok, it took me about 30 mins to come up with a road plan to calculate this out like mathematician would, bow it would take me ~5 whole minutes to actually calculate it! Luckily, I can simply spend an entire weekend coding a script to do it for me!

@ScrotN 3 месяца назад

“My students can see my mistakes when teaching” I am that student

@MathTheWorld 3 месяца назад

Are you saying that you are the kind of student that can see when professors make mistakes? Or are you saying that you are one of my particular students that has pointed out my errors in class before? Great job for being either one, but if you're the latter, you can testify to all that I make a lot of mistakes.

@justgreg8756 3 месяца назад

My dad would always optimize it by grabbing a spare drink lid and filing one of those up with sauce.

@TarenNauxen 3 месяца назад

I do this too, just make sure to put a napkin under the straw hole!

@bumpty9830 3 месяца назад

Your dad would've been a great engineer.

@ineedhelp8573 2 месяца назад

I put my fries on tissue and use the container for it. Trust me it works

@tiripoulain 3 месяца назад

My gut tells me that 45 degrees is optimal if the base has a 0 radius, whereas in the limit as the base gets really wide, the optimal angle ought to approach 0 degrees

@arctan4547 3 месяца назад

thats two times the area of two triangles right? so maximize cos(theta)sin(theta) one multiplication rule later we get cos(theta)(cos(theta))-sin(theta)sin(theta) angle addition formulas say we get cos(2 * theta) thats pi/4 for the intercept, you were correct, and I survived self doubt of doing all of this while not even having taken a geometry class

@jestercab42 3 месяца назад

If the base was 0 then the corresponding shape would be a regular cone and so we can just use the formula pi*base*height/3 and writing it in terms of theta with the given 2.5 cm of the side we have the function (pi*(2.5*sin(theta))^2)(2.5*cos(theta))/3. Then if we optimize this we actually get an angle of about 54 degrees and not 45.

@jestercab42 3 месяца назад

@@arctan4547 I don't understand what you mean by two times the area of two triangles. Isn't the case of a 0 radius base just a circular base cone?

@MikehMike01 3 месяца назад

@@jestercab42equilateral triangle has the most area so if the base is zero I would expect 60°

@alexanderbudianto7794 3 месяца назад

I decided to do the calculation myself with an arbitrary radius and slant height. Turns out, it really depends on the ratio between the radius and slant height, and the exact solution is very complicated, involving the arcsine of the cubic formula. As the radius approaches 0 (which makes it a cone), the optimal angle approaches arcsin(sqrt(2/3))=54.74° and the maximum volume if the slant height is 1 is 2sqrt(3)pi/27 (though deriving this is much easier by treating it as a cone to begin with; using the aforementioned formula would require using complex numbers as well).

@schizoframia4874 3 месяца назад

It’s funny how a problem so mundane can be solved with calculus. Love it

@Fire_Axus 3 месяца назад

your feelings are irrational

@bumpty9830 3 месяца назад

Calculus (and really, differential equations) is more fundamental to daily experience than we tend to realize. For example, the basic relationship between position, speed and acceleration, which we understand intuitively enough to throw baseballs accurately, involves a second-order differential equation. The American education system treats calculus like rocket science, but that's a feature of the culture rather than a truth about mathematics.

@v0id_d3m0n Месяц назад

@@bumpty9830 yeah basic calculus itself is so easy... I guess the troubling thing for people might be how abstract it gets and how it's like a gateway to the complex world of pure mathematics and advanced calculus.

@donniemorrow 3 месяца назад

Field Manual for the Practical Saucer: Optimal sauce fillage can be achieved by yanking apart alternate pleats in the cup to achieve a uniform 45° angle. Yank every 3rd pleat if you need a good balance of sturdiness and volume without risking sauce spillage, especially if the sauce is liquidy or you are on-the-go

@MathTheWorld 3 месяца назад

@donniemorrow Thank you for your post. We have had a lot of math and physics in the comments, but you have added some vital engineering. Well Done!

@francescodenaro6081 3 месяца назад

Best channel on YT.

@MathTheWorld 3 месяца назад

😭😭😭

@fspar 3 месяца назад

A hundred percent.

@SmokeyChipOatley 3 месяца назад

Totally agree. Informative and wholesome to boot.

@FireyDeath4 3 месяца назад

Every time I read a remark to this effect, it makes me sad about how much all the different people saying this about different channels miss out on, and how we'll never see how they get to respond to other people saying it But Of course We all Know #sweetheartthebest Roflololol

@Imhappy440 3 месяца назад

...........

@MasterHigure 3 месяца назад

"No one has thought about before"? Excuse me, but not only have I thought about this. I have *taught* this in my class. And also I use it regularly when filling my ketchup cups at Micky D's. I remember being ecstatic the first time I realized I didn't need two cups to get enough ketchup to last me through all my fries. Taking into account the non-Newtonianness of ketchup and considering that the surface doesn't have to be flat and level will certainly push the optimum in the direction of flat-and-wide. My gut says "by a lot".

@DragTheDecimator 3 месяца назад

surprisingly, my Calc2 professor mentioned a similar optimization problem when we were first discussing the topic. It wasn't specifically sauce cups, but trapezoidal bowls, and also about volume. we never did solve it in-lecture though.

@keywestalert6329 3 месяца назад

He did the math, and he could afford 8 kids plus himself.

@ghosttwo2 3 месяца назад

I actually worked this exact problem around 2006 when I was doing calculus homework at the Arby's I was working at. It's funny because I saw the ketchup and knew what it would be.

@MathTheWorld 3 месяца назад

Wow! I'm so impressed! How many people are doing calculus while working at Arby's?

@aldomorelli6379 3 месяца назад

Although we cant use the fundamental theorem of calculus when differentiating the integral at 7:30, we can use the Leibniz Integral Rule/Differentiation Under the Integral Sign to still take this derivative without having to calculate the original integral first.

@0x4849 3 месяца назад

Commenting on 7:36: you _can_ apply the fundamental theorem of calculus when differentiating with respect to a different variable as integrating. So: d/dx int_{0}^{x} f(t) dt = f(x). The reason this doesn't work here, while the bound is a function of theta, is because theta is also the variable of integration, and not independent.

@viktorsmets29 3 месяца назад

Yeah y is just a dummy variable, you can call it whatever you want, it doesn't matter outside the integral.

@someguyontheinternet9186 3 месяца назад

Couldn’t you do a funny little change of variables to fix this?

@mycoffee2654 3 месяца назад

This channel is the best! It's just a little above my math skill, so I feel like I'm learning a lot every time something comes out! It also applies to real life questions with intuitive explanations! Thank you!

@MathTheWorld 3 месяца назад

You are welcome! We love the positive feedback!

@manuelb__r 3 месяца назад

An chemistry and physics overkill would be to calculate the optimized shape in order to use less paper and take advantage of the viscosity of the sauce to allow it to get over the top without overflowing/spilling

@jessehammer123 3 месяца назад

I watched silently as you went through the frustum strategy, then when you mentioned doing it via calculus I audibly muttered to myself “that’s how I’d do it!” That was fun.

@Kaptitanbear 3 месяца назад

nobody has ever asked this question, but i also haven't ever seen your channel before, and yet both are enrapturing me with their specificity

@error.418 3 месяца назад

I have asked and solved this question in my own way like a decade ago

@darring1070 3 месяца назад

I never thought that trying to figure out how much sauce goes in a cup would break my brain.

@RenTheFen 3 месяца назад

Math is for sure my favorite subject. When I was in high school, it was easily my favorite class. I liked how you could learn so much from math and be able to apply it virtually everywhere. It is incredibly fascinating and satisfying! Watching this was a blast :)

@-jes988 3 месяца назад

After being presented with the task, my intuition was, maybe a bit less than 45°. Feels good to be so spot on 😊🎉

@paradiselost9946 3 месяца назад

i see it as two curves. at 45 degrees, the volume of the conical section has the max volume, rising from zero at either extreme. whereas the cylinder simply increases in volume with height. by bringing the height slightly above 45, the cylinder gains more volume than the conical section loses. and i came up at around 40 degrees being best... to write it as an equation? lol...

@Aurea-Highdiocratis 2 месяца назад

Mrs.E showed this to us in class recently, I am just gonna say I am so amazed by how mathematicians can apply abstract content into real life. Your video made these abstract concept fun to watch!

@MathTheWorld 2 месяца назад

Thank you! I am glad it was fun for you. I am also soooo happy that teachers/instructors/profs are using this in class. That is the dream for my channel is that they can be used to help students see the power math has to make sense of the world around us. Good Luck in Class! (I am actually amazed on the other side, how so many mathematicians tend to take such applicable/powerful mathematics and make it so abstract/disconnected from the real-world! )

@MisterLobb 3 месяца назад

I think there is another factor to consider: the objective isn’t to find optimal volume the cup can theoretically contain, but how you can deliver to the table in a single cup. When it’s unfolded, it loses rigidity in the walls and when you lift it, the force of the fingers deforming the walls will cause waste. I think you will find that the cup manufacturer took this into account in determining the size of the folds so the original shape and size maximizes deliverable volume then adjusts down for the sake of safety. So the actual max volume will slightly larger than the manufacturer’s fold size.

@MathTheWorld 3 месяца назад

Thanks for bringing in the engineering aspect. That is something I haven't taken into account (and don't know the science enough to actually do so).

@MisterLobb 3 месяца назад

@@MathTheWorld Yeah, it was out of the scope of your experiment. I just pointed out that the design of the cups - for example; the paper quality/thickness, folds & wax coating are determined by the manufacturer’s engineers to get an optimal rigidity for cost. Things we take for granted often have a lot of thought behind them. Just like your experiment. Hopefully exposure to your channel inspires kids to pursue these interests into meaningful careers

@LeventK 3 месяца назад

You have a unique and engaging style, never change it! It works.

@MathTheWorld 3 месяца назад

Thank you!

@keaganhurter2550 3 месяца назад

btw, i love your channel, i always get so excited when you upload a new video, they're just such great explanations

@MathTheWorld 3 месяца назад

Thank you so much!

@yaoitiddieexpert1486 3 месяца назад

well. you could also account for the amount of sauce you can stack over the top of the shape, which would actually favour a wider top to give it more space to spill out

@MathTheWorld 3 месяца назад

This is what we give us the challenge problem at the end of the video. It's worth extra nerd stickers if you solve it!

@canis_lupus2220 3 месяца назад

@@MathTheWorld This is when a "simple" math problem turns into a complex physics question. What is the viscosity of the sauce? At what pressure does it exit the sauce dispenser?

@mujtabaalam5907 3 месяца назад

Now model the % overlap due to folding and accout for that in the unfolding. There's also no need for the volume to be a frustum; you can use the surface area of the unfolded paper and variational calculus to find the shape that holds the most volume (accounting for the lump at the top with the surface tension, which you could model as as a lopped off sphere with some experimentally derived contact angle).

@someguyontheinternet9186 3 месяца назад

Now this is pod racing

@otavionunes6756 3 месяца назад

hello, i'm very curious about this problem, would you mind elaborating on how this can be done? I will make an attempt myself.

@voidify3 3 месяца назад

The fact that your son has 7 siblings wasn’t even relevant to the story

@RaffieFaffie Месяц назад

He's flexing

@acursedhope 3 месяца назад

That title is on point, made a fairly trivial problem interesting enough to click on Great video all around, keep it up!! I love practical calculations

@OmnipotentEntity 3 месяца назад

I 100% have asked this question as a kid. Before I knew about calculus. My dad never finished high school, so I never actually figured it out. Thanks for satisfying my ancient nostalgic question that I never quite got back around to. You're an awesome dad.

@MathTheWorld 3 месяца назад

Thanks! My son Spencer loved that I made a video that came from his quest for maximum sauce.

@er4795 3 месяца назад

You can't apply FTC at 7:38 but you can apply the Reynolds Transport Theorem!!! Also known as differentiation under the integral sign. For those who don't know, there are two issues here. 1) The variables with respect to which we are integrating and differentiating are different. In fact the integrand depends on both variables! 2) The bounds of the integral vary with the derivitative integral. Reynolds Transport Theorem in 1 dimension solves this and yields a formula in terms of another integral and the net flux through the bounds.

@victor-oh 3 месяца назад

I've been thinking about this optimization problem to overkill with math too. What is the optimal time period to refuel your car, given inflation and cashback parameters? Cashback is when I get a weekly limit of how much money I get discounted from my purchases. I use a complete tank every two weeks.

@MathTheWorld 3 месяца назад

great question! we'll add it to our list of potential future videos!

@dougcorey3830 3 месяца назад

That might depend on how far out of the way you have to go for gas, since that will increase the cost of refueling.

@victor-oh 3 месяца назад

@@dougcorey3830 assume fuel station is within commute route

@mycoffee2654 3 месяца назад

That sounds like a great video idea!

@marivexx 3 месяца назад

i feel less alone in my love of solving math problems just to solve them for funzies

@MathTheWorld 3 месяца назад

There's a whole field of math called recreational mathematics just for people like us. Martin Gardner is the king of writing about recreational mathematics. Another nice place to start is the mathematical puzzles of Sam Loyd.

@Mythical_Myths16 3 месяца назад

Ngl same

@mrsheldon9134 3 месяца назад

This is just a simplified version of one of my final exam questions when I was 16, but with Arby's cups instead of a random generic container.

@MathTheWorld 3 месяца назад

I am impressed, because I don't think I could have handled this question as a student without some guidance.

@justinbrentwood1299 2 месяца назад

Having recently taken an introductory analysis class, I have something to add. One could consider the volume of the ketchup as a function of the angle at which the cup is extended up from the ground, where the domain is the closed set from 0 to 90 degrees. Since closed intervals are compact sets (that's a separate proof but trust me), and this function is continuous by observation when comparing it with the definition, we know that the function has a maximum and that it attains its maximum. That is, we can justify that this problem even has a solution.

@fridging 2 месяца назад

god tier shitposting

@jonasls 3 месяца назад

Bros optimizing for maximum kids though

@jcorey333 3 месяца назад

Cool video! As a fellow lover of Arby's sauce, I'll keep this in mind 🤤.

@thedarkcity4084 3 месяца назад

As someone learning Optimization for the first time, I feel like I would do something like this in the future out of boredom

@eragonawesome 3 месяца назад

This is my favorite kind of math video, where you learn a whole bunch for a silly reason

@MathTheWorld 3 месяца назад

Thanks! We enjoyed making it, and plan on making more. I do like the ones where someone learns a lot and learns something about the world as well (a not-so-silly reason), which is what most of our videos are..

@christiansrensen8330 3 месяца назад

Brilliant! I love it. Thank you for using metric. It is appreciated :)

@MathTheWorld 3 месяца назад

The people have been heard

@MXvsATVnovice 3 месяца назад

Bro 8 kids dawg

@MathTheWorld 3 месяца назад

Thanks!

@c0d3w4rri0r 3 месяца назад

A more interesting extension would be to drop the assumption that you have to have a conic section and a flat base. The paper could theoretically be folded into any surface of revolution provided the cross-section had a line of the appropriate length. So as a problem consider using the calculus of variations to find the surface of revolution that for a particular radius Circular piece of paper gives the maximum volume.

@MathTheWorld 3 месяца назад

You're right, that is a really nice extension! Now you have me thinking of doing this as a follow-up video.

@otavionunes6756 3 месяца назад

@@MathTheWorld please do I am very interested, your channel is great!

@Ptaku93 3 месяца назад

I drifted off around the 4 minute mark, but congratulations on having 8 children!

@annika6081 3 месяца назад

We have the maths.

@icantchooseaname6903 2 месяца назад

your wife must really love math..

@Airbigbawls 3 месяца назад

Your handwriting is extremely satisfying. I know nothing about math and got lost after the first formula was written. Stayed for the mesmerizing handwriting.

@MathTheWorld 3 месяца назад

Wow what a compliment thank you 😭

@37_tranhoangtuan73 3 месяца назад

hey I love your idea, but for me, technically, to get more sauce, just open it all the way, and overfill, OVERFILL the disk to the edge hehe, I always do that, but I sure love the vibe here

@aircraftape 3 месяца назад

48 seconds since the upload of the video lol

@MathTheWorld 3 месяца назад

a true fan!

@CoolAsFreya 3 месяца назад

This sounds like exactly the kind of inconsequential math problem I'd focus on after ordering fast food

@MathTheWorld 3 месяца назад

Yes! Earn that nerd sticker!

@SwankiestPants 3 месяца назад

Pro strat (and likely a much harder calculation) you can bow out the sides of those paper cups without changing the diameter of the rim allowing for much higher volume of sauce

@MathTheWorld 3 месяца назад

I actually think the optimal would be somewhere in between your and mine, where the rim is increased, but you aren't restricted to having straight sides. Stay tuned.

@dexterouslaboratory 2 месяца назад

When I was young, I would do something like this, but I hadn’t come across the “hack” to open the paper cup. My mind came up with what I knew about increasing volume - blowing up a balloon. By placing the mouth of the cup against my mouth, placing my hand flat on the bottom of the cup and inflating the cup, it would get larger. I postulate that this method is more efficient. The height is affected minimally, but the surface area is expanded as the pleats unfold into a sphere-ish shape maximizing volume for a given and increased surface area.

@BirnieMac1 3 месяца назад

We don’t have arby’s here, but spencer’s plight is important to me (because optimisation and his enjoyment)

@fluminox13 3 месяца назад

That math was pretty harmless. And it is never overkill to save a bit of money. Nice video

@dillonstrange5152 3 месяца назад

Enjoyed watching this, then saw the BYU logo at the end and got really excited as I am a current BYU student. It's nice to see professors putting out content like this, keep it up!

@MathTheWorld 3 месяца назад

Come by my office sometime. I would love to meet you. 171A TMCB.

@voidify3 3 месяца назад

Ah, mormons, that explains the 8 kids

@noodles24601 3 месяца назад

Another reason expanding the cups can be useful is that the wider rim and corners make it easier to get every last bit of sauce out onto your food. In their default state you often end up with either very shallow dips at the end or a lot left in the bottom.

@Amphvptere 3 месяца назад

The derivative of this man’s kid count over time is fucking insane

@mirihawk 3 месяца назад

I’d calculate the instantaneous rate of this guy’s fertility but I’m avoiding studying for my calc exam in 3 days

@Amphvptere 3 месяца назад

@@mirihawk haha real, just finished my exams, good luck

@pi_xi 3 месяца назад

I never thought about the idea to calculate the volume of a truncated cone with an integral, but it works.

@nearautomatons6987 3 месяца назад

An interesting thing about these paper cups is that it’s often possible to spread out the sides *without* breaking the circular rim creating a more pot like shape. This way you can expand out the sides without decreasing the height and still maintaining most of the structural integrity. I don’t know if it’s *mathematically* optimal, but in terms of actual use it’s pretty close

@yolandaconitzer7333 2 месяца назад

I adore this! Somehow I can hear your explanation but not my teachers, when teaching the same thing. Also this less than useful maths problems are great

@midflight_art 2 месяца назад

as someone who also grew up on Arby's (and these paper cups) I always pinched the tops but pushed out the middles of the wall for these cups so it looked like a swollen barrel without a top.

@thepeterbro1 3 месяца назад

You sound like the coolest professor ever, there needs to be more teachers like you!

@MathTheWorld 3 месяца назад

Thank you! I don't know if my students learn much more math in my class, but I think we do have more fun!

@TheAnantaSesa 3 месяца назад

My first guess was 45° angle was optimal just bc of the triangle areas that form on the sides. Glad to know that math supports my hunch.

@2020_Gaming 2 месяца назад

I 3D modeled and am currently printing one of these for the cups I have at home! Gonna have a nice little holder to keep them optimal!

@fruglemonkey 3 месяца назад

I've always had a similar question - when you dip a fry into sauce, the level of the sauce rises in the cup, increasing the area of fry covered by sauce per dip distance. IE, if you had a sauce container that was only slightly wider than the width of a fry, you wouldn't have to push the fry into said container very far to cover it heavily with sauce. What would the optimum container shape be to minimise the dip distance required to cover a quarter of a fry with sauce, while still holding enough sauce to be able to be used for multiple fry dips?

@PopeGoliath 3 месяца назад

Totally flat! Dip sideways. Never have to go much more than a fry-width deep.

@Comp_Laments 3 месяца назад

This is the greatest math video ive ever watched. Ive never found out more useful info from any video ever than what I found here

@MathTheWorld 3 месяца назад

Amazing thank you!

@rainytreecat3992 3 месяца назад

I once looked at a cup of ketchup and thought about this exact problem. I thought "Yeah this is definitely possible" then never thought about it again

@nikk-named Месяц назад

I am mourning the inner math nerd in me, given how little I understand of this. Love the maths overkill regardless. It's fun!

@MathTheWorld Месяц назад

We're glad you enjoyed the video!

@motive-li1do 2 месяца назад

For a very long time i read this title as "the math optimization problem but no one cares about my son" and i thought it was about some well known word problem solved by not caring about the needs of one of the people mentioned in it

@keaganhurter2550 3 месяца назад

idk if this is correct. but my proposal to finding the optimal angel is simply taking the original equation for the frustum and adding a cylinder on top. this cylinder would have the volume pi*(r^2)*S where S is the height of the sauce. plunging in the value for r we get ((Lsin(theta)+R)/2)*pi*S which, when adding to the original formula we get find that that the optimal angel is 0.732 radians

@peccantis 3 месяца назад

My field work suggests this advanced manipulation method is easily superior: spread the bottom out as much as it gives, but allow the rim to remain largely tucked Maximum sauce AND maximum stability.

@OLLGY69 2 месяца назад

4:00 If you're the short side of a right side triangle, consider yourself an opp

@powderedphantom5765 3 месяца назад

This sounds like one of those extreme scenario maths problems you get given in primary school.

@slowitty8918 3 месяца назад

I do this with the cups, and i have actually started the math, then just been like "I'll just grab more cups", lol. So many thanks for this answer to my laziness

@37_tranhoangtuan73 3 месяца назад

love this overkill man, sometimes I want RU-vid to be filled with videos like this and people like you

@_yuri 3 месяца назад

"who is perhaps 15" , a classic dad moment when father doesnt remember the age or school grade of child.

@Athakaspen 3 месяца назад

I definitely pondered this problem in middle school or so

@trueriver1950 3 месяца назад

You don't need the trig. By Pythagoras: r = R + sqrt ( (2.5)² - h² ) Then use that to eliminate r from the volume formula. Find where the derivative of the volume is zero. Check we got a max.

@MathTheWorld 3 месяца назад

Yep! I just think it's easier to think about angles in the situation. I also thought a lot of viewers might struggle to understand the non trig functions.

@marcfoster2260 3 месяца назад

I was preparing to complain that this solution was not quite overkill since we had not considered the angle of repose for this sauce, but you hinted at addressing this near the end of the video.

@dgaf9843 3 месяца назад

I was yelling sauce viscosity at my phone for 8 mins and you had to pull out those last ten seconds.

@hippolippo 3 месяца назад

You said that you guarantee no one has asked this but I was literally thinking about this yesterday, thank you for solving this now I can maximize how much ham I get in my omelet at the dining hall

@kamo7293 2 месяца назад

1:25 that's the answer that gets "boooos" from the math crowd 😂

@MathTheWorld 2 месяца назад

Exactly right!

@aterxter3437 3 месяца назад

About the cup's volume there is a last option which is the second pappus-guldik theorem : The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (The centroid of F is usually different from the centroid of its boundary curve C.) Seems impossible at first sight, but can easily be prooved for the triangle and rectangle, then with the linearity of integrals, any surface work

@MathTheWorld 3 месяца назад

Very nice! You're right, I could have used this one. I teach a generalization of this theorem to my students, but I use it less with volumes and more to do with work and hydrostatic Force problems. They love it!

@Pystro 3 месяца назад

Makes you wonder why the manufacturer of the cups doesn't maximize the volume. (It's probably because the optimized cup would be much wider and a stack of these cups would take up more volume. And putting that much sauce into the cup and then spreading it out into a plate for dipping can easily make it overflow.) But if the manufacturer wanted to maximize the volume, they could also change how much of the circular piece of paper they leave for the bottom.

@MathTheWorld 3 месяца назад

There's another good reason why they don't maximize the volume with their design. They're trying to keep people from taking too much sauce. The more sauce they take is money out of their pocket.

@NotKnafo 2 месяца назад

all you need to to figure it out is to have as many glasses as there are folds, then pour water to the cup then empty it into one of the glasses and unfold one fold repeat that until you got all of them the fullest glass will correspond to the the optimized cup

@MathTheWorld 2 месяца назад

You can do a similar strategy without the glasses. Some other commenters have suggested filling the cup to the brim, undo one pleat, if the level dropped add more sauce up to the rim, undo a pleat, etc. until when you went to a fleet the sauce level doesn't fall.

@HelPfeffer 3 месяца назад

0:04 I was in a mathematics competition as a kid once, where one of the questions wondered how to fold a carton so the box made out of it would have the most volume under certain constraints which feels like the exact same problem for a mathematician

@robertstevensii4018 3 месяца назад

"This is clearly math overkill" -Famous last words before someone goes down what's likely the greatest rabbithole in the universe

@MathTheWorld 3 месяца назад

If you think this is bad, just wait until the follow-up video.

@vinijoncrafts2882 3 месяца назад

another awnser I thought about is to observe that the 2d cross section of the cup consists of a square then two congruent right triangles. So you can basically bring one of the triangles to complete a rectangle with the other. So the problem is now to find the biggest area the whole rectangle can make, if that makes sense, which is much easier.

@spoonmanxx 3 месяца назад

It would not be a math overkill!

@helleye311 3 месяца назад

I did in fact think about it before. But i never really got as far as actually doing the math. I just assumed it's 45° or thereabouts. Glad to see i was pretty close.