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?

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!

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.

My sleep deprived brain is telling me that there's an easier way of finding this out with way less math Just bring a little scale w u to Arby's and get a bunch of those little cups and open them to various capacities, full them up then get an empty cup to zero out the scale and and measure the cups The heaviest contains the prime sauce (I'm writing this before I finish the video- if he talks ab doing it this way I'll retract my statement :D)

0:53 you misunderstand, acceleration as a stat isn’t that important, it’s mini turbo that is important. Mini turbo is a hidden stat that usually corresponds with acceleration, and changes how long your drift boosts last

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

Please there has been a problem that has been atuck on my mind for 1 year. Draw 1 right isosceles triangle. Draw a line in the middle of it, splitting it into 2 triangles. If we keep splitting all the triangle, then how much triangle would it produce? And what is the pattern to it?

And then u get reminded about the surface tension of the sauce and realize how useless highschool math is

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

Love the video 😂, though normally mario cart players max their mini boost stat! ❤

"I've scaled the court so that units are feet" - my rage meter jumps from 0 to high. "But we will put the metrics measurements on the screen as well" - my rage meter drops from high to 0.

I care.

Would have love if same analysis was applied to sports with weight classes like wrestling to see if there are body types that excel even when weight is controlled

Please consider cropping out the flashing text in the bottom right. It’s all I can focus on when in shot

42 is the answer

Tldw:41 degré

U could do better video about integration.

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.

Couldn't you just treat this as a 2D problem and solve for maximum area?

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

I think the speed climbing route is similarly arbitrary, although I think technique can conteract that more than it can in hurdles

The hidden stat called mini-turbo is arguably the most important stat in the game

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

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.

just discovered your videos , you are one of the small bunch that can make math fun !!!

Wholesome and intelligent, thank you

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?

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.

"Adding up tiny bits" seemed like the most realistic interpretation, infinitesimals notwithstanding.

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.

Next: instead of Math Overkill, its Math Slaughterhouse

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.

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

7:45 gambling mb?

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

This is how our teachers expect us to use math

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.

More like math oversauce.

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

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)

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.

I wish I saw this on Sunday, could have helped me with my final exam

This is a lesson in how good math can lead to wrong conclusions due to missing variables. There is a hidden stat that influences how fast a mini turbo charges up, and low speed/high acceleration carts usually have a better turbo stat than fast carts. Counterintuitively this means that slow carts in practice are actually faster when nothing is going wrong. It's the low acceleration carts that are better at catching up because their high speed gets multiplied by power ups (mushrooms, stars, etc) that are only available when you're losing.

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

I personally think that you're trying to gaslight us into thinking you get laid when this whole video is an obviously testament to how you clearly don't.

The next problem sounds like a physics problem

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

Homie did calculus to find out yoshi teddy was op

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

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