The Optimization Problem No One Cares About But My Son
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- Опубликовано: 8 май 2024
- 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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Created by Doug Corey
Script: Doug Corey and Jennifer Canizales
Audio: Doug Corey
Animation: Jennifer Canizales
Music: Coma Media
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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: ruclips.net/video/jSR-QR2GRqQ/видео.html
How many of the creases do I need to unfold for that angle?
that all is a great theoretical answer, but practically how many folds do you have to untuck to get closest to the optimal angle?
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.
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
8 KIDS???? IN THIS ECONOMY?
Free sauce is free sauce
When you know math, you can afford it apparently
He knows math don't question him😅
@@itsyo42which sauce are you referring?
@@gideonk123 mayo
8 kids? bro keeping the fertility rates UP on his own
the optimal technique
Bro found the optimal impregnation time slot, with 8 nerd stickers to boot
Practising multiplication
7 boys and 1 girl. Can you imagine? that's like a 0.1% chance
@@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%.
Wait 8 kids, you are the math problem guy
He must have calculated it's Cheaper By the Dozen
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.
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%.
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.
may I have the codes if it still available? I'm still new to python and im just figuring things out.
@@Nano-n Sure thing! I'll add some better commentary and send it when I get home today.
@@HawkulusQuest thank you for the reply
8 kids💀💀💀bros trying to make a math clan
A math class.
Families in math problems be like
Bro IS the math problem
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
I think it's quite common for olympiads to have these types of questions.
@@Kokurorokukoolympiads having calculus LMAOOOO you innocent soul
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.
@@magicmeatball4013 what?
@@Kokurorokuko the math Olympiad has 0 calculus on it, it’s innocent to think that scary calc problems would appear on olympiad level stuff
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.
In fact it is very likely asked and answered intensively or it's very hard on higher levels
Why human wanna be unique ? 🤔
Bc otherwise we're too easy to replace @@setusof
@@seeker296you say that like it’s a guarantee
@@alex.g7317its close to one
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
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
@@siliconhawk9293 you should have tried to fit the word domain in a few more times in that comment 😂😂
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.
You are welcome! Do you have any other problems that have stumped you?
@@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?
“My students can see my mistakes when teaching”
I am that student
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.
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!!
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)
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.
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.
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.
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!
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...
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.
@@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...
@@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.
@@malvoliosfWith ketchup being a non-Newtonian fluid (seriously) I think treating it as a collection of particles might still be applicable!
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...)
The fact that your son has 7 siblings wasn’t even relevant to the story
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
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
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.
@@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?
@@jestercab42equilateral triangle has the most area so if the base is zero I would expect 60°
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).
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!
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.
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.
Thank you we agree!
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.
ruclips.net/video/g4bNhXX1oRw/видео.html - I made this video in which I did just that!
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.
It’s funny how a problem so mundane can be solved with calculus. Love it
your feelings are irrational
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.
He did the math, and he could afford 8 kids plus himself.
My dad would always optimize it by grabbing a spare drink lid and filing one of those up with sauce.
I do this too, just make sure to put a napkin under the straw hole!
Your dad would've been a great engineer.
I put my fries on tissue and use the container for it. Trust me it works
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
@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!
Best channel on YT.
😭😭😭
A hundred percent.
Totally agree. Informative and wholesome to boot.
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
...........
"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".
I never thought that trying to figure out how much sauce goes in a cup would break my brain.
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.
Wow! I'm so impressed! How many people are doing calculus while working at Arby's?
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.
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
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!
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!
You are welcome! We love the positive feedback!
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 :)
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.
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.
Yeah y is just a dummy variable, you can call it whatever you want, it doesn't matter outside the integral.
Couldn’t you do a funny little change of variables to fix this?
You have a unique and engaging style, never change it! It works.
Thank you!
This is something I thought about a few times years ago, back when this would be mentioned as a thing to do in life hack videos, but I never really fully thought about it because I don't really eat sauces when I go to fast food places with these paper cups. I'm glad this video came up in my feed though because I've given you a subscribe.
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.
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
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
I have asked and solved this question in my own way like a decade ago
btw, i love your channel, i always get so excited when you upload a new video, they're just such great explanations
Thank you so much!
oh dude love this, immediately it came to me as a classic related rates problem you learn in your first math semester! great video, was quite fun
Brilliant! I love it. Thank you for using metric. It is appreciated :)
The people have been heard
Bro 8 kids dawg
Thanks!
This is what i watch RUclips for! Great great work. Thank you for taking the time to make this kind of content 🎉
You are welcome! I'm glad you enjoyed it.
You are welcome! I am glad you enjoyed it.
I dont know if this is really on topic, but this warmed my heart after seeing some really nasty things on reddit. best of luck to you, you seem like a good father.
After being presented with the task, my intuition was, maybe a bit less than 45°. Feels good to be so spot on 😊🎉
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...
Cool video! As a fellow lover of Arby's sauce, I'll keep this in mind 🤤.
I remember i asked myself this question a lot, thank you for the explanation
Great question! We'll add it to our list of potential future videos!
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.
i feel less alone in my love of solving math problems just to solve them for funzies
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.
Ngl same
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.
great question! we'll add it to our list of potential future videos!
That might depend on how far out of the way you have to go for gas, since that will increase the cost of refueling.
@@dougcorey3830 assume fuel station is within commute route
That sounds like a great video idea!
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.
Thanks! My son Spencer loved that I made a video that came from his quest for maximum sauce.
Thank you for making this video
My pleasure!
We have the maths.
48 seconds since the upload of the video lol
a true fan!
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
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!
@@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?
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
Amazing thank you!
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.
You sound like the coolest professor ever, there needs to be more teachers like you!
Thank you! I don't know if my students learn much more math in my class, but I think we do have more fun!
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
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.
You've done it again math the world man!
Thanks! I'm glad you enjoyed the video!
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.
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).
@@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
Your handwriting is extremely satisfying. I know nothing about math and got lost after the first formula was written. Stayed for the mesmerizing handwriting.
Wow what a compliment thank you 😭
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.
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.
Ive wondered this before thanks
You are welcome!
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!
Come by my office sometime. I would love to meet you. 171A TMCB.
Ah, mormons, that explains the 8 kids
I never thought about the idea to calculate the volume of a truncated cone with an integral, but it works.
wow your son must be pretty cool for coming up with this question!
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
I definitely pondered this problem in middle school or so
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).
Now this is pod racing
hello, i'm very curious about this problem, would you mind elaborating on how this can be done? I will make an attempt myself.
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
I drifted off around the 4 minute mark, but congratulations on having 8 children!
I'd be really interested to see a video showing how some equations were dirived. Like how Einstein's work with clocks influenced how he approached the math that ultimately helped lead him to GR.
What was that process like? What does it take to create an equation that uses analogies to express complex properties that you then need to create equations that describe complex behaviors with high precision.
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.
As a math and sauce enthusiast, this was right up my alley. But as a rule bender aligned chatotic neutral, I know that I get the most sauce (as well as a bigger/better surface area for dipping) by just using a drink lid 🤣
This sounds like exactly the kind of inconsequential math problem I'd focus on after ordering fast food
Yes! Earn that nerd sticker!
love this overkill man, sometimes I want RUclips to be filled with videos like this and people like you
As someone learning Optimization for the first time, I feel like I would do something like this in the future out of boredom
Wonderful video
Fellow math nerd here. Love this.
Hahaha ... great stuff. Subscribed. I look forward to your other videos. Cheers ...
Thanks! We appreciate it!
We don’t have arby’s here, but spencer’s plight is important to me (because optimisation and his enjoyment)
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
This is my favorite kind of math video, where you learn a whole bunch for a silly reason
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..
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.
You're right, that is a really nice extension! Now you have me thinking of doing this as a follow-up video.
@@MathTheWorld please do I am very interested, your channel is great!
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.
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
That math was pretty harmless. And it is never overkill to save a bit of money. Nice video
The actual trick is to get a large drink cup lid, and put it in there. Just trust me on this one, mate.
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
My numerical methods teacher gave us a similar problem and my approach was very similar than yours. It was a piece of steel and bend it to make a drain (idk how to call it other than that). The only data we were given was the width of the sheet (I think it was 20 cm) and told to optimize the total area of that trough numerical methods.
Really fun! to find out the max height we can go above the cup we could also find out density, viscosity and surface tension data for the specific type of sauce, and then use this to find the maximum shape before the cup overflows. i believe this is way more accurate than the height=5mm approximation you mention. Also, this video assumes that the sides of the cup remain as straight lines, which is not necessarily true. How could we adapt this method to do so?
I guessed it would be where sin&cos meet at .7071, the top corners arc down decreasing by Cos() so the increasing Sin() to balance optimally it would work out where they meet, Im happy it works out to be a pretty close estimate though
I believe, you actually can apply derivative in the second solution though. Leibnitz condition is satisfied, so you can interchange integral and derivative in this case and (maybe, 'cause I haven't tried it yet) make calculations a little bit easier. Not in the FTC sense, obviously, but still, in come cases it may help
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.
NOW THIS IS A GOOD USE OF MATH
Thanks! But I think any use of math is a good use.
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
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.
Really fun problem and I now have a page full of diagrams of frustums (frusta?), trying to find the optimal solution for an arbitrary bottom radius and side length. Eventually I decided "none of these are going to produce an *elegant* solution, I'll keep playing with it later."
I love the idea of using a solid of revolution formula! One caveat: the reason we couldn't use the FTC right away in that case is NOT because we're integrating wrt to y, and differentiating wrt theta. You'd ALWAYS be integrating with respect to a different variable, because it's a bound variable. You then differentiate with respect to the upper bound of the integral. In this case, the upper bound is a function of theta, rather than just theta, but that's not a problem--the chain rule can be applied. The real problem is that a function of theta appears as a coefficient within the integrand!
To solve this integral explicitly, I'd expand the square in the integrand and break the integral into a sum of three, factoring out the coefficient tan(theta) from the integrals as needed. Then when I differentiate, I can use the product rule for these terms, and the FTC to differentiate the integrals. It's too fiddly a process to really be called *elegant* in my opinion, but it should work!
Do we get into physics and calculate max above rim for various viscosities? Do we assume various flexibilities in the cup and maximize WITHOUT collapsing the cup under the weight of the sauce?