Can You Solve The Martini Glass Puzzle? A Simple Illusion That Fools Most People

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Just how full is your cocktail glass anyway? The answer will surprise you.
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21 май 2024

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Комментарии : 854

@webbraham2768 Месяц назад

I am half way through the video and forgot about the third dimension

@nemoyatpeace Месяц назад

Yep, me too.

@randomname9291 Месяц назад

That was my exact mistake

@aba_dab_o Месяц назад

Same. 😅 Was thinking between 70% and 80%, but closer to 70%.

@JonnyBoi957 Месяц назад

Yea I guessed that is what most people though. That is why I guessed 80% as it was a cone.

@jonothanthrace1530 Месяц назад

But first, we need to talk about parallel universes.

@trimeta Месяц назад

Part of why so many people's intuition was 70% is that in the 2D case, the answer would be 1/sqrt(2), which is 70.7%. And although the wording was very clear that we're interested in the 3D case, the pictures themselves are naturally 2D, which colors our intuition.

@yfxxiii Месяц назад

Yeah, my brain was doing maths with triangles because that was the visual I was presented with, instead of cones. It's another fun way to think about how easy to the mind is to trick (and often does the tricking itself).

@StRanGerManY Месяц назад

Pictures are not naturally 2d. They are artificially 2d to mislead and confuse.

@trimeta Месяц назад

@@StRanGerManY What's the shape of the screen you're viewing this on?

@talkingbirb2808 Месяц назад

@@trimeta you don't know how to draw a cone on a piece of paper? edit: I took a closer look and it's drawn like a 3D cone

@trimeta Месяц назад

@@talkingbirb2808 Without doing a full-on animation (which arguably does add a third dimension -- time), it's hard to draw a cone that doesn't look like a triangle.

@47shawty12 Месяц назад

as a previous bartender, i knew it was 80% instantly. if you were to pour it into a glass of the same volume but with a cylindrical shape, it would be at the halfway mark.

@spacewolfjr Месяц назад

another round, please

@CollegeHustler Месяц назад

I was going to say the same thing! lol Real world experience is priceless!

@user-sl7ie9te5r Месяц назад

@@spacewolfjr another martini, Paul?

@abhishankpaul Месяц назад

What if a mathematician shows up there?

@phoquenahol7245 28 дней назад

@@abhishankpaul 2^(-1/3)

@deemjeffrey Месяц назад

I guessed 80% bc i’m a bartender. Literally no calculation, just from working with martinis. Then you started proving it with math and I got so confused haha. Love your content. Thanks for everything!

@AiNaKa 29 дней назад

i'm not a bartender and i didn't calculate it but still guessed it right, i just figured it was common sense given the understanding that thinner volumes hold less fluid than wider volumes. martinis are cone shaped, so i figured they'd exaggerate the effect quite a bit, so my initial guess was 80%

@LONKULADE 29 дней назад

@@AiNaKa yea that's what I thought

@Zinozad 27 дней назад

I guessed 80% because it looked like that was the closest to half full. Not a bartender or anything.

@colinjava8447 26 дней назад

Technically 100/2^(1/3)% = 79.37% The angle doesn't matter cause stretching the glass horizontally and depth-wise will preserve the proportions.

@BiglerSakura День назад

Just imagined a situation: I ask a bartender to bring me 2 full glasses of martini. He brings 2 grasses seemingly almost full and a bill charging for 2 glasses. In front of him, I pour all the martini from one glass into the other, and the level of the liquid still doesn't reach the edge of the glass. So I say I'm gonna pay only for 1 glass. Will this work in a real bar?

@Bleaksigilkeep 29 дней назад

As a former bartender I do have to make a minor correction in that a Martini glass actually should be filled to nearly the rim. The purpose of the glass is to give the volatile aromatic molecules a large surface area to evaporate into the air but not to enclose and capture them, so that when sipping you get a strong aroma from the drink only at the moment you first bring it to your face to sip, not throughout the motion of tipping the glass to drink. The glass should be filled almost completely to the rim, like 95%. Even 80 or 90 will give too much room for the aromas to collect and the intended effect of the first sip will be list, as the drinks nose will be full of hot alcohols and terpenes

@zinaidalogunova5731 Месяц назад

Cone is really good shape of glasses for bars’ owners😀

@gregoryt1139 Месяц назад

And cubes. Don't forget cubes...Ice cubes. Plenty, plenty ice cubes.

@no_mnom Месяц назад

If they don't fill it to the top, it's a sham!

@_xano Месяц назад

„Con” is literally in the cone name so

@Clyntax 26 дней назад

A much simpler way to approach this is to observe that both cones are similar. So you want to scale down the larger cone in order to half its volume. Scaling a body does not depend on the shape at all, it can even be done with a cube. If you scale a body in 3D by factor s in every dimension, the volume increases by a factor of s^3. We want to know the scale factor x so that the volume factor is one half: 0.5 = s^3. Therefore, x = 0.5^(1/3) which is almost 0.8 and the answer to the question. No pi, radius, graphs or complicated formulas needed.

@andredavis4657 13 дней назад

That's the way I did it

@Zeptonixmusic 21 день назад

I think that 80% of the height = 50% of the volume is not as mindblowing as 20% of the height being 0.8% of the volume

@piershanson1784 Месяц назад

One thing about eyballing it is that when doing the poll with just eyballing it, I am naturally inclined to pick the triangle where the orange area is half of the volume rather than remembering that the triangle represents a cone which is what we're actually supposed to eyeball. When I eyball the triangle, 70% is definitely closer to half the area of a triangle, but since the triangle is supposed to be a cone, he 80% actually wins out.

@paulgreen9059 Месяц назад

Without using a calculator I realized the answer was the cube root of one half. Then I needed a calculator.

@chrisglosser7318 Месяц назад

I had it memorized from when I used to to teach physics

@thomasdalton1508 Месяц назад

I approximately cubed 1.2 and 1.3 (I figured working with the reciprocals was easier - that may or may not have been true) in my head, realised it had to be somewhere between them and guessed 1.25, which corresponds to 80% so went with that.

@ThatFoxxoLeo Месяц назад

Given the initial question was multi-choice, you could've worked backwards from there. 80% is 4/5, so you can just multiply 4/5 by itself twice to get its cube (4/5, 16/25, 64/125). 64/125 is very close to 1/2.

@christopherwellman2364 Месяц назад

@@ThatFoxxoLeoI like that explanation. Thank you.

@Dreamprism Месяц назад

512 is 8^3, so .512 is .8^3, so .8 is close to cbrt(.5)

@wmpowell8 Месяц назад

There's an intuitive explanation for the % volume = (% height)^3 formula: as the glass is filled, the shape that the water makes is scaled larger and larger with the tip of the cone anchored in place. Since a cone is three-dimensional, the volume of the cone is scaled in accordance with the cube of the scale factor of the lengths, thus, % volume = (% height)^3.

@martianunlimited Месяц назад

I just used pretty much that, r is proportional to h , let's call r = ah, so rewriting that we have V = 1/3 a^2 h^3 since a is a constant, for V' to be 1/2 V, h' would just be 1/2^(1/3) h; it feels like we are just overthinking the problem.

@stuchly1 Месяц назад

@@martianunlimitedbut both of you are doing calculations 😂

@1104Tea Месяц назад

Its easy to assume wrong when you're presented with a 2-d image for the options, in a question that wants an answer based on 3 dimensions. What everyone learns in school is to go with what information is presented if the problem doesn't specify any detail. I know some people will try to be cheeky and say they top of the drawing may imply something, but we all know that can just be there as an artists choice for making any generic 2d cup.

@Tomyb15 Месяц назад

Yeah, I didn't think about the diagrams enough and made a simple mental calculation based on 2D cups and got 70%, but in 3D it gives 80%.

@senbatifanola Месяц назад

Ah yes a 2D glass

@Questerer Месяц назад

I’ve never heard of a 2D liquid. The question implies that it is in 3D.

@Misteribel Месяц назад

I'd love you to pour me a 2d drink! How many ml go into a 2d martini glass? 😂

@SuperClavera Месяц назад

@@Questerer When 60% assumes something "wrong" based on how the question is presented, then there's absolutely something wrong with the question, especially since their answer is correct when calculating the area in 2D.

@rorywquin Месяц назад

I knew the answer immediately. I came across this in the early 1980s. I was in a bar (in a place called Klerksdorp in South Africa) when two people ordered liqueurs. One wanted a single and the other wanted a double. The barmaid poured their drinks (by eye) into the same size glass. The guy with the double complained because he felt the guy with the single was getting a better deal (more than a single). She got a tot measure & poured a single into it and topped the glass (with the single) up. It came to exactly the same level as the double (she was pretty good at her job).

@MrMousley 20 дней назад

Exactly the same thing happend to me when I worked behind a bar, A 'double' served in a proper martini glass is NOT 'twice the height of liquid in the glass'. That's why I always used a measure and poured the drink into the glass in front of the customer.

@neuralwarp Месяц назад

All the glasses are 100% full. It's just a matter of what they're full of.

@user-pr6ed3ri2k Месяц назад

my glass is filled with 90% water and 10% bose-einstein condensate

@Ejemplo-lz8ql Месяц назад

"It's just a MATTER of what they're full of" Pun intended?

@fatmccat1513 Месяц назад

@@user-pr6ed3ri2k Mine has 10% quantum foam

@user-pr6ed3ri2k Месяц назад

@@fatmccat1513 what's the other 90%

@Inspirator_AG112 Месяц назад

*Remember that the real-life counterpart of that diagram is 3-dimensional...*

@Nukestarmaster Месяц назад

Yeah, that was what got me, lol. 2d diagrams are a nasty trick.

@SteinGauslaaStrindhaug 29 дней назад

1:20 Assuming the inside is not a truncated cone, i.e. it's pointy all the way down, the angle should not matter. My intuition says it's around 75%, but it wouldn't surprise me if it's closer to 80%.

@StephanBuchin Месяц назад

I'm even more impressed by the fact that the 50% height is only 12.5% full.

@justdilka Месяц назад

I'm a martini glass half full kinda person

@Minetendo_Fan Месяц назад

So are you 1/8 full or 80% full?`

@Matty0311MMS Месяц назад

Maybe @justdilka is in a quantum superposition of both?

@chobies5383 27 дней назад

I'm a " Martini to the half way point"

@suntzupup Месяц назад

I knew it was 80. Didn't calculate it but I could feel it when I pour and it had pissed me off for years.

@kyyzh12 Месяц назад

I remember me and my dad figuring out where you would have to cut a cone into 2 perfect pieces. We also got ~79.4%

@noomade Месяц назад

surely you cut the cone vertically down the middle....

@kyyzh12 29 дней назад

@@noomade lol this is what we thought when seperating a cone into 3 equal pieces

@bootlegharold6696 Месяц назад

Presh that was needlessly overcomplicated lol

@ThePowerfulOne07 Месяц назад

That explains why my iPhone thinks 80% battery level is considered full!!!

@Temporary_yesyes Месяц назад

it only goes to 80% to preserve battery life overall

@ThePowerfulOne07 Месяц назад

@@Temporary_yesyes trust I know and for full performance and capacity as well!

@amanavinash-fb4zk Месяц назад

this video explains the opposite

@chriswebster24 Месяц назад

If the height of the liquid is 80%, the glass would only be about half full. If the glass is 80% full, the height of the liquid would be about 93%.

@sunrevolver Месяц назад

Rofl

@maxc300s Месяц назад

People that clicked 70 percent because they thought of the martini as 2d and not 3d here 👇👇👇👇👇👇

@annoyingbstard9407 22 дня назад

Did you really do that? 😂

@jwdory 4 дня назад

I never heard of a 2d martini. What are you a flat lander?😅

@henk-ottolimburg7947 Месяц назад

You don't need the formula of a cone, for many very difficult with pi etc. The glass can be any form It's enough if the partly filled glass is an image of the full glass. The factor is 0.6 or 0.7 and it works in 3 dimension to the 3rd power. 0.8 * 0.8 * 0.8 is approx 0.5

@frumbert Месяц назад

INtuitively I said 80%, since the volume goes up the wider the glass. Nice to see it explained.

@Dexaan Месяц назад

This is why there's usually something like (20 cl) next to the wine descriptions in restaurants. Mixed drinks are usually sold by the shot, as well.

@hippophile Месяц назад

70% makes sense for our eyes because we intuitively see 70% as half the glass, looking at it we see a 2-dimensional picture - and indeed, the area of the smaller triangle at 70% height is 49%, because it is a square relationship. Cubic relationship is more extreme, so 80% feels right...

@ShotgunLlama Месяц назад

Now hol up. Who said it has to be a cone? What if we're looking at a cross-section of a triangular prism?

@deathpacito8702 Месяц назад

The math works out to be the same, since volume still scales with height^3 EDIT: NVM, mixed up prism and pyramid. If it's a triangular prism then yeah 70% should be the answer here.

@duanecjohnson Месяц назад

Doesn't make a difference. Any prism is 1/3 cubed route of height. A cone is just a many-sided prism. AD0TJ

@geraldsmith6225 Месяц назад

Than it wouldn't be a martin glass

@protoman1365 Месяц назад

@@deathpacito8702the volume of a rectangular prism is triangular area (which is the cross section we see) * the length of the glass in the Z axis. Since the length of the glass at any given moment is unchanged, wouldn’t the answer change to 70%, as only two dimensions change instead of three and it’s related to the square of the height instead of the cube?

@deathpacito8702 Месяц назад

@@protoman1365 Ah my b, mixed up prism and pyramid

@musanim 12 дней назад

Or to put it another way: the liquid in the glass is a 3-dimensional volume in which the x, y, and z sizes (height, width, depth) all vary proportionally together. Volume is x*y*z, so, the answer is the cube root of one half.

@alopexlagopus1488 Месяц назад

As soon as percentages were introduced without the x100 factor when one side of the equation was cubed, it makes the equation out by a factor of x10000. Very surprised at Presh for incorrectly converting to percentages rather than sticking with ratios.

@Lost_City007 Месяц назад

8:03 @MindYourDecisions He said " v/V = % of the volume of the large cone ". No, it is only the percentage like ( y %). Value with %... So the relation is y % = (x %)³

@kylejacobs1247 Месяц назад

The misleading part of this question is that it is presented as a 2D problem visually, when in fact it is a 3D problem.

@harry2.01 Месяц назад

I fell into the same trap.

@MrJoerT Месяц назад

In 2d 80% is still the closest answer, right?

@lidarman2 Месяц назад

@@MrJoerT I think it is 70.7% for 2d case. Solve the 2d version, 1/2 = x^2.

@c.jishnu378 Месяц назад

Underrated.

@drenz1523 Месяц назад

Visually 2d? There's an oval at the top, an oval at the bottom, glasses irl are clearly 3 dimensional, 2d glasses don't exist (closest you'll get is a very flat prism glass), etc

@oneeyejack2 Месяц назад

If two 3D objects are similar, the ratio of the volumes is always the ratio of lengths cubed

@r.markclayton4821 Месяц назад

@lefthanded3512 Месяц назад

Wow, surface area really makes a difference

@Nfscarbon07 Месяц назад

YES I figured 80% because of videos I've seen about shot glasses showing how much alcohol you can lose out on, 70 was my first guess but it seemed low to me

@romasgaucas 12 дней назад

This problem was on today's (June 10th) Lithuanian National Mathematics exam. It was on the test part. The volume of the cone is 1350 ml. How many ml are there in the cone if it is filled to 2/3 of the height. A 400 ml B 600 ml C 700 ml D 900 ml

@petersmythe6462 Месяц назад

Note that the exact taper of the glass matters. Any self-similar shape, that is, shapes which maintain their relationship to a cylinder containing them as they fill, will fill with an exponent in accordance with its fractional volume of a cylinder. A cylinder will fill linearly. A paraboloid will fill quadratically A cone will fill cubically. A long tapered shape taking only a quarter the volume of its bounding cylinder will fill quadratically.

@bpark10001 Месяц назад

The angle of the taper DOES NOT MATTER as long as the sides are straight.

@nilton61 Месяц назад

A cone implies straight sides

@katrinabryce Месяц назад

It doesn't matter what the angle of taper is, what matters is that it is 0 at the bottom, and goes linearly to some positive value at the top.

@suhaasvemuri7980 Месяц назад

Pretty easy question if you learned introductory calculus. The main point to know is that the radius of a cone is proportional to the height on the slant of the cone (so at 50 percent height, the radius would also be 50 percent of the top). The formua for a cone is 1/3 pi r^2 h, and lets take out the constant 1/3 pi since it really doesn't matter. If we set an arbitrary value for radius and height (lets say 1), the full volume is 1^2 x 1 = 1. Now, just keep plugging in values until you get a volume equal to half the full volume, or 1/2. 0.8^2 x 0.8 = 0.512, so 80 percent is the closest.

@khaitomretro Месяц назад

Pretty easy question if you've done some cooking and have used a Tala measuring cone for ingredients.

@suhaasvemuri7980 Месяц назад

@@khaitomretro lol that too

@Dejiek0 Месяц назад

I approached it with assuming the cross sections perpendicular to the y-axis were circles. The radius of the circle is found using the base of similar triangles so then the volume can be easily found by integrating along the y-axis. Then I kept plugging in values until I found what the closest to 50% is. Pretty fun little problem, but you're right, this is single variable integral calculus.

@meta02 Месяц назад

Well yeah but the question specifically said to not calculate it for this reason.

@Dejiek0 Месяц назад

@@meta02 Correct. I guessed 80% before going on with the calculations.

@rogue5882 Месяц назад

Without having to rearrange, the volume scale factor for a 3d object is (the length scale factor) ^3

@chimingchan9038 Месяц назад

Very nice analysis👍👍

@fullfungo Месяц назад

There is also a nice intuition even if you don’t know the exact formula. If we scale a 3-D shape in all directions by a factor of S, then any length measurement L will be scaled by S (for example, the height will scale proportionally); any areas measurement A will scale by S•S (for example, the base); and any volume measurement V will scale by S•S•S (for example, the whole volume). So a martini glass filled up to 50% (1/2) of the height (L) will have a volume (V) of 1/2•1/2•1/2 = 1/8. But if we scale the height by 80% (0.8), the volume will scale by 0.8•0.8•0.8 = 0.512. The same math works for cubes, spheres, cylinders, ducks, houses, ice and any other 3-D objects.

@Dejiek0 Месяц назад

Very interesting. I solved this with integration using similar triangles before I saw your elegant solution.

@dcterr1 29 дней назад

i paused this video at 1:34. It's obvious to me that the correct answer is 80%, since (0.8)^3 = 0.512 is the closest of the given values to 50%. Note that the shape of the liquid in the half-full glass is similar to the shape of the full glass regardless of the slope of the glass, whence the constant of proportionality of the linear dimensions of the full glass to the half full glass is the cube root of 2, or approximately 1.26.

@lidarman2 Месяц назад

I knew the answer because I recalled doing a similar thing in 1st semester calculus 40 years ago. I think we had to calculate the rate of volume change vs height of liquid change. I also thought of the pizza size vs diameter optimization here and knew since it was 3d, it would be a cubic instead of a square.

@RR-vk2tl Месяц назад

That is the reason why they use this shape of glass in the bars

@drelijahmikail3916 Месяц назад

we can `sense` the intuition with the volume formula: V = pi*r^2*h/3, that the r is a power of 2 with delta{h}. Therein, delta{h} in increment of %, can lead to a power of 2 increase.

@lordsmeagol3390 29 дней назад

I immediately concluded: volume of cone => 1/3 base area times height; Base area is proportional to the square of the height => volume proportional to height cubed => for 1/2 volume, height is cube root of 1/2

@MorgurEdits Месяц назад

If you have one of those measuring containers with similar shape you can notice how often the value changes at the top and how rarely it changes on the bottom, that is why I was one of the 23%

@dennisdesormier6886 28 дней назад

A simpler version of the math: At a fraction p of the height, the radius is the same fraction p of the max radius because of the similar triangles. The volume for fraction p is V(p) = ⅓ π (pr)² (ph) = ⅓ π r² h • p³ = Total volume • p³. So if you want 50% volume, you need p³ = 0.5, and that's where p ≈ 80% comes from.

@umchoyka Месяц назад

80% - I had a good intuition on this. As someone who makes a pot of pour over coffee every single morning in a cone filter, I have a very keen awareness of when the pot is almost done brewing.

@mattjanzen2344 Месяц назад

Bartender here. Drink recipes are measured independent of the glass. Then you consider "wash lines" - the point at which a given glass is visually "full", while still allowing it to be carried comfortably without spilling. With final volume of the drink and volumes of various glasses at their wash lines all known, an appropriate glass can be selected - or recipe and price adjusted.

@duanecjohnson Месяц назад

A cone is a special form of a pyramid. Which is also the cubed route of base * height. AD0TJ

@nzeches Месяц назад

Height is proportional to the radius, so volume is proportional to the cube of height. Hence 1/2=v2/v1 = (h2/h1)^3

@Bavarianscience Месяц назад

There's actually a shorter way to do the calculation. The cross sectional area of the cone shaped glass is equal to pi*r^2. r is directly proportional to the hight because it's a cone so the area has to be directly proportional to the hight squared. Then we can simply integrate this h^2 term with respect to the h and we get the volume to be directly proportional to 1/3*h^3. The factor of 1/3 can be ignored like any other constant factors thus far. That way it becomes clear that the answer has to be close to the cube root of 1/2 which is 79,37% or nearly 80%.

@johnrains8409 16 дней назад

Before anyone asks, whether a glass is half full or half empty (assuming it has liquud up to the halfway point) depends on the sign of derivative of the level with respect to time, dL/dt. To determine this, you must know the level just prior to it reaching the halfway point. If this is deri ati e is negative, the level is going down and it is half empty. If it is positive, the level is rising and the glass is half full.

@sugurlu1 Месяц назад

Great video, thank you.

@JayTemple 22 дня назад

As a side note, you're right that the angle doesn't affect the answer to the question at hand. What it affects is just how much it holds. If two glasses have the same height from point to rim but differing widths, the volume will vary directly as the square of the width. So, if the question had been whether you come closest to 50% of the volume with one whose rim is 90%, 80%, 70% or 50% of the original, the answer would indeed be 70%.

@gaijininja Месяц назад

Damn, I was one of the small percentage that guessed 90% on the earlier poll. I thought it would be similar to the question about lily pads covering a lake by doubling each day. (That one is 1/2 covered on the second to last day.)

@angrytedtalks 29 дней назад

Definitely more than 70% but less than 80%. The mind trick is to stop thinking of it as a flat image. If it were a triangle, the half height would be 25% the _area_ of the triangle, rather than 12.5% the _volume_ of the glass. In 3D this is exaggerated so volume becomes more distorted compared with the visual assumption.

@Altoclarinets Месяц назад

I sat down and mathed this out for the fun of it, but if you asked me this question at a party, I would just get a book (or other hard, flat object), firmly cover the end of the glass, and turn it horizontal to see which one had a liquid level that reached the point of the cone. This is called solving by brute force, and while it's not mathematically efficient when there are many examples to be tested, for a set as small as this (particularly considering that I had already eyeballed the 80% one as looking right so probably would have began there) it is good for making your point quite aggressively

@williamlennie Месяц назад

Solving this problem is way more fun with trig!

@WhoStoleMyAlias Месяц назад

I'm sure that a math teacher would applaud this approach, but there is a much simpler way to find the answer. The thing to realize here is that because of the triangular shape you can express the height of the fluid level as a linear function of the surface radius, i.e. h=c·r or the other way around r=c·h. Yes I know what you want to say but bear with me. Given this relation we can express the volume of the fluid as a linear function of h³, or the reverse 'h' as a linear function of the cubic root of the volume. Thus if you want to decrease the volume with some factor 'x' this means that 'h' must decrease by a factor equal to the cubic root of 'x'. Also note that the glass does not need to be round for this logic to apply. It works the exact same way if the glass is shaped octagonally as long as each possible cross section maintains the triangular shape.

@MicMan123456789 Месяц назад

80% seemed right to me just because the area of the crossections increase by a squared factor

@KiranRajagopalanMusic 29 дней назад

Cuemath is doing a wonderful job, thanks for calling it out 💪

@randomname9291 Месяц назад

My answer was 70% but I think it’s because I assumed the glass was 2d. This was my calculation Using similar triangles, we know that the ratio of surface areas between the drink and the glass is equal to the ratio of their heights squared, which is (7/10)^2 (70% out of 100%) meaning the surface area ratio between the drink and the glass is 49/100, which is the closest value out of the 4 to 50/100=1/2. However, since the glass is 3d, we have to actually raise the power to 3. (7/10)^3=343/1000=34.3% Whereas (8/10)^3 =64/125=51.2% I think most people chose 70% because it looks the closest to halfway full in the little drawing which is true since the drawing is 2d

@eskewroberts7663 17 часов назад

Before watching, I'm gonna say 80% This is a volume question, and the radius and height of the liquid will increase or decrease at the same rate. Because we're dealing in 3 dimensions, the volume will decrease exponentially as the height and radius decrease, and 80% just kinda feels right

@jean-francoisbouzereau6258 Месяц назад

In the graph, the axes must only be oriented toward the positive direction only.

@markbernier8434 Месяц назад

Anyone who messes about in boats knows this. Most fuel gauges are basically rulers. You quickly learn that if you go below 3/4 on the dial you can't get back. (more than 50% fuel is expended) There are lots of very carefully hand constructed graphs in chart tables that show what you did by formulae.

@feynthefallen 23 дня назад

I saw this demonstrated in tv education program back when I was a child. They did it for several shapes and demonstrated the relation of shape to certain graph shapes. Taught me more about mathematics than any two of my school teachers.

@vbinsider 24 дня назад

The only thing I knew for sure right at the beginning was that the answer does neither depend on the angle nor on the volume of the glass. It's basically the same result as for why the water pressure does not depend on the area of the sea in which you are diving but only on the diving depth.

@trentonrothan9724 Месяц назад

I know why most people picked the wrong answer. The 2D representation almost looks like a triangle. Given the visual depiction, 70% full looks like it would be half full as would be the case with a triangle. The picture is misleading compared to an actual martini glass, thus most people got it wrong. When comparing the empty space on the 2D model that is empty and full 70% looks like the right answer.

@djwarner7144 Месяц назад

High school geometry says the area of similar figures vary with the square of the ratio of linear dimensions. Similarly, the volume of similar solid figures vary with the cube of the ratio of linear dimensions.

@Matty0311MMS Месяц назад

I answered this intuitively, and calculated it later by cubing the decimal values of the percentages. I picked 80%.

@thegreatgreenarkleseizure1994 Месяц назад

I did the math 'wrong' in that, in my head, I knew the volume was 1/3*area*height, and I just thought that I would multiply that by 1/2 being the 50% height mark, but intuitively the halves aren't 'equal' like they would be in a cylinder, but more volume in the widest (upper half). Thus roughly 1/6 (1/3*1/2) of the 'fat' side of the cone height is where the 50% should be and that is about 17%, thus 83% 'full' is halfway there going from the other direction. Not correct, but a close estimate without dealing with roots and whatnot. Glad you pointed out my error (and I always love this stuff because I've been out of school for decades).

@shruggzdastr8-facedclown 28 дней назад

This feels related to the cube-square law, which demonstrates that there is a lag between how much the surface-area of a given solid increases when you double its volume. Said surface-area only increases by the square of the cube-rt of 2 (e.g.: 2^(2/3)). Conversely, when you double the given solid's surface-area, you more than double its volume

@Fred-yq3fs Месяц назад

Very easy problem. The volume of the glass at any height is Base*h/3 Base = Pi.r^2 Because of triangle similarity (tangent formula works just as well), we have r = r0(h/h0) where r0 is the radius of the full glass and r the radius at height h. Therefore V/V0 = (h/h0)^3 cube 0.6 0.216 0.7 0.343 0.8 0.512

@sebwiers1 Месяц назад

You don't need to know how to calculate the volume of a cone at all to answer the question. Simply consider that when partly filled, the filled portion is a scaled down version of the full glass. Volume scales as the cube of a liner dimension. Cubing .80 is closer to .50 than any of the other options. Yes, doing the calculations proves that, but bypassing the calculations with an intuitive understanding of the relationship between scale and volume is what the 28% who got it right did. The 60% who said the 70% height was closer were almost certainly going off the relationship of scale to area, if just subconsciously.

@GroovingPict 29 дней назад

you say it's completely counterintuitive, but I not only got the correct answer, but also figured it was slightly overshooting (ie that the 80% height was slightly more than 50% volume) which was also correct. No calculation (I wouldnt know how to do that anyway), just intuition

@aerbon 29 дней назад

i like to do these just off of the thumbnail. instinctively, i was torn between 70 and 80% with some quick mental math: scaling the height of the liquid here is scaling the volume of liquid in all 3 dimensions. 0.8^3 = .512 so the 80% glass is 51.2% full. in 2d 70% is the correct answer as 0.7^2 = 0.49 so i don't blame the brain for guessing 70% when looking at a flat image.

@markfoxwell2411 Месяц назад

A good reason for using calibrated shot measures.

@Maxime-fo8iv Месяц назад

I usually solve your riddles from the miniature, then I just skip to the answer and like. Not sure if that helps the algorithm, but in any case thank you very much for all those riddles!

@jensraab2902 Месяц назад

I think the easier approach to calculating this is to use the intercept theorem. If you scale up (or down) any geometric object any given area will change with the square of the scaling factor (because areas are two-dimensional) and any given volume will change with the cube of the scaling factor (because volumes are three-dimensional). This is because all measures scale up (or down) with that same scaling factor. Because, if held upright, a cone standing on its head (as is the case with this Martini glass) filled up by x% will result in another, similar cone, the volume of this cone of liquid will be (x%)³. All that said, even though I knew this, my first impulse was to say 70%. It really doesn't look like two 80%-full glasses would (almost) fit into one full glass!

@billjohnson3858 Месяц назад

The area of the cross-section at any height is proportional to the square of the height. The volume swept by the cross-sections from the bottom to the height is the integral of the area vs height function, so it is proportional to the cube of the height. If the height of the top is 1, the volume could be represented as 1*1*1*K = K (K is just the actual volume of the glass). Similarly, the volume at 50% would be 0.5*0.5*0.5*K = 0.125K, which is 12.5% of K. If we want the height that is 1/2 of the volume, we need the cube root of 0.5 which is 0.793701 (or 79.3701% of the height).

@marcvanleeuwen5986 29 дней назад

Would you consider the following cheating (by doing a calculation)? In music, one third of an octave is (in an equal tempered scale) a major third, and a pure major third is a frequency ratio of 4:5. So the answer must be about 4/5=80%. The initial "third" come from drinks being in dimension 3, and the octave is a ratio 1:2 for a half-full glass. The "third" in "major third" has nothing to do with this (and a minor third is in fact one fourth of an octave).

@wintersxlstice21 Месяц назад

i’ll keep this in mind for my bladder

@user-pr6ed3ri2k Месяц назад

So, it's an upside down cone? I've never seen one of these, but here we go... Volume of a cone = πr²h/3 Assuming r = 1 and h = 2, the volume here is 2π/3. π/3 would be the half volume. The radius can be expressed from the height with the function r=h/2 (which I guess can be changed depending on the ratios) so πh³/12 = 2π/3 would indeed lead to the original h = 2 π/3 instead of 2π/3 would lead to h = ³√4? The start of the video seems to imply we should be using the angle, but like, how are you planning to measure that? The radius and the height are much easier. EDIT: 4:06 h=1 r=1/2 and h=2 r=1 does indeed result in this effect, π/12 vs 2π/3 is a difference of 1/8 aka 12.5%. 7:38 Yeah, just change this formula to use different ratios and stuff. 8:23 oh crap. Didn't expect that.

@0dd701 Месяц назад

just find out which of the values of 0.6^3,0.7^3,0.8^3 and ).9^3 is closest to 0.5. so in 0.216,0.343,0.576 and 0.729, 0.576 is closest to 0.5 , the answer is 80%

@weevilinabox Месяц назад

5:49 Isn't a "right-angled cone" one return a side angle of 45⁰, i.e. where base diameter equals height. I believe that what is discussed here is a "right cone", i.e. one whose tip is is perpendicularly above the centre of the base circle. Happy to be corrected. (Disclaimer: I'm in the UK. Terminology may differ across the pond.)

@doctorb9264 Месяц назад

Excellent problem and solution.

@billh5923 Месяц назад

Off the top of my head guessing, volume is proportional to height times radius squared and radius is proportional to height, so volume is proportional to height cubed. 0.8^3 is about 1/2. So answer is 80%

@tomas.caklos 18 дней назад

You don’t need cone formula nor any complex calculation. The cones are by principle similar and volumes of any similar objects differ by cubes of their height. Just count 0.8^3 and get it.

@JLvatron Месяц назад

Math for Alcoholics! lol!

@jensraab2902 Месяц назад

Hey, if that's what it takes to make people interested in math, so be it! You know those people who whine about math, saying "when will I ever need this in real life?" Well, here's an example!

@gorak9000 Месяц назад

iDz tHe BezZt kInDuh mAtHhHhHhs

@JLvatron Месяц назад

TeeHee-ous replies!

@mathewmunro3770 29 дней назад

Doubling the height cubes the volume (height, radius, radius all go up proportionally to height), so just ask whether 0.8^3 is closer to 0.5 than 0.7^3 is, and the answer is yes, much closer.

@SiqueScarface Месяц назад

I would indeed do the calculations immediately - being an engineer. But on the other hand - a volume scales up with the third power, hence an 80% full glass is quite a decent guess, as 0.8³ is 0.512.

@Queenside_Rook Месяц назад

Without watching, unlike most solids, filling a cone from the point up is equivalent to scaling it along a single dimension, so with the square cube law, if you want 1/2 the volume, you need to scale by (1/2)^(1/3) ~= .794 So 80%

@BeyondKawaii Месяц назад

What I did first is taking the formula of the volume and substituted r with its expression through h and the angle. Now that h became the only variable - I solved for the ratio of h, which would half the volume.

@Matthew.Sweeney 23 дня назад

very sneaky to make it look like a poll about area when it's really a poll about volume

@X22GJP Месяц назад

My intuition was correct - just looked right; but then I immediately thought in terms of volume.

@niello5944 Месяц назад

The reason so many people answer. 70% instead of 80% is because the images shown are in 2D. Had it been 3D more people would think about how it's a circle that's getting filled each layer.