The Optimization Problem No One Cares About But My Son

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.

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?”

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Created by Doug Corey

Script: Doug Corey and Jennifer Canizales
Audio: Doug Corey
Animation: Jennifer Canizales
Music: Coma Media

© 2023 BYU

[FinleyRichard]

Wait 8 kids, you are the math problem guy

[HawkulusQuest]

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.

[yanniammari]

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

[rithishkr]

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

[Funnymoney]

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.

[abduking.]

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

[ben_]

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

[HagalazI]

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.

[malvoliosf]

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...

[dbarcene]

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

[eurosenelsuelo]

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.

[donniemorrow]

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

[tiripoulain]

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

[schizoframia]

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

[OLLGY]

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

[cappincrrunch]

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!

[justgreg]

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

[x]

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.

[aldomorelli]

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.

[DragTheDecimator]

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.