Very few people can solve. Which martini glass is closest to 1/2 full?

[@sh4wty7]

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]

another round, please

[@CollegeHustler]

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

[@ДмитрийОсипов-м9д]

​@@spacewolfjr another martini, Paul?

[@comrade_marshal]

What if a mathematician shows up there?

[@phoquenahol7245]

@@comrade_marshal 2^(-1/3)

[@trimeta]

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]

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]

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

[@trimeta]

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

[@talkingbirb2808]

@@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]

@@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.

[@Zeptonixmusic]

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

[@maze7050]

Reverse Pareto principle lmao

[@Nudtanun]

really?

[@Nudtanun]

omg i use calc and it real

[@stanislavmedvedev9377]

Well, Pareto is usually about 20% and 80% of smth or vice versa. Not 20% and 0.8%.

[@maze7050]

@@stanislavmedvedev9377 Yes, that's why I said it's like "REVERSE" Pareto principle; instead of 20% holding 80%, 20% holds 0.8%.

[@webbraham2768]

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

[@nemoyatpeace]

Yep, me too.

[@randomname9291]

That was my exact mistake

[@aba_dab_o]

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

[@JonnyBoi957]

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

[@jonothanthrace1530]

But first, we need to talk about parallel universes.

[@StephanBuchin]

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

[@jasonward9429]

Which is why you 1)Make sure the bar man/maid uses a measure.2) if there is no measure involved, the liquor should go in BEFORE the ice.

[@TirthBhesaniya]

10% is 0.1% full

[@deemjeffrey]

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]

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]

​@@AiNaKa yea that's what I thought

[@Zinozad]

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

[@colinjava8447]

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]

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?

[@piershanson1784]

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.

[@wmpowell8]

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]

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]

​@@martianunlimitedbut both of you are doing calculations 😂

[@feierhou3731]

I think of it in a unit conversion way: volume has cm^3 as its unit, height has cm as its unit, so 1/2 volume should mean height^3 = 1/2, and 0.8^3 is the closest to 1/2. However, I was still doing calculations, and without calculations, I won’t be able to guess.

[@BillyViBritannia]

​@@stuchly1how much is 2+2? Is it 4 or 6? You are not allowed to do any calculations.

[@cyrion7819]

r³ because in every dimension it expands linearly. Like a sphere or a cube. r² goes with things that expand in two dimensions like areas oder cylinders.

[@SteinGauslaaStrindhaug]

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%.

[@paulgreen9059]

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

[@chrisglosser7318]

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

[@thomasdalton1508]

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.

[@BaeYeou]

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]

​@@BaeYeouI like that explanation. Thank you.

[@Dreamprism]

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

[@TruthSurge]

1:18 this is a volume determination and I THINK you need calculus to figure that out properly. 70& LOOKS the closest to me but maybe 80% is the real answer. I don't know but I know that 50% of the height is NOT the correct answer. :)

[@laufert7100]

If by needing calculus you mean to figure out the volume of a cone yes. If instead you know the volume of the cone beforehand, the algebra is quite simple and results in the cube root of 1/2

[@freedomhawk772]

This is one of the best, if the the best, of all the math channels on KZbin. I appreciate the accuracy of Mind Your Decisions.

[@zinaidalogunova5731]

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

[@gregoryt1139]

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

[@no_mnom]

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

[@xanowich]

„Con” is literally in the cone name so

[@BillyViBritannia]

​@@gregoryt1139martini on the rocks it is...

[@denvercheddie]

Actually there is nothing wrong with the intuition that says 70%. That's because the pictures are in 2D. A triangle would need to be filled 66.7% to be half full. If we saw the glass in 3D, our intuition might be different.

[@oggyoggy1299]

There are no triangle glasses.

[@denvercheddie]

@@oggyoggy1299 I know that, but the comment is about perception. When solving the math problem, we are not thinking glasses, we are thinking triangles and cones.

[@panteleymonschekochikhin-k1978]

That's not right, triangle filled to 66.7% of height is not half full, it's 44% full.

[@denvercheddie]

@@panteleymonschekochikhin-k1978 You're right. Silly error on my part. It's 0.707...

[@1104Tea]

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]

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]

Ah yes a 2D glass

[@Questerer]

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

[@Misteribel]

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

[@SuperClavera]

@@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.

[@Inspirator_AG112]

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

[@Nukestarmaster]

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

[@castleanthrax1833]

The number of comments (on the poll. Not this video) who were saying that it depends on whether the glass is 3d or 2d, astounded me. How anyone could fill a 2d glass with liquid is beyond my understanding.

[@mofprailes7140]

Can work tho with 2d drawing animation I guess

[@castleanthrax1833]

@@mofprailes7140 The question was clear. It said LIQUID.

[@Songfugel]

Because they forgot what the problem was, since the graphic is 2D with a very hard to see 3D glass rim, especially on mobile. It is pretty easy mistake to make if not paying attention

[@GooogleGoglee]

In 2 dimensions I can color the surface of the glass with a marker.... That is liquid. But hey 😂 I am joking here 😉

[@Idiomatick]

My guess was based on 2d as well... i just saw it as a triangle

[@kyyzh12]

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%

[@kyyzh12]

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

[@stigcc]

I would like to have a cone cut in three equal volume parts, all representing one of the conic sections. Is that possible to make?

[@PinkeySuavo]

@kyyzh12 where did u cut it though? if you cut in 80% of height you won't get equal pieces anyways. The best would be to cut through the height

[@kyyzh12]

@@PinkeySuavo well no duh you can cut it lengthwise. it was just a fun math problem to try

[@neuralwarp]

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

[@user-pr6ed3ri2k]

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

[@fatmccat1513]

@@user-pr6ed3ri2k Mine has 10% quantum foam

[@user-pr6ed3ri2k]

@@fatmccat1513 what's the other 90%

[@qorymij37]

​@@user-pr6ed3ri2k plasma made of 71% nitrogen, 28% oxygen 1% argon

[@mc3941]

This is peak optimism

[@Clyntax]

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]

That's the way I did it

[@PinkeySuavo]

but still u wont do 0.5^(1/3) in ur head :P

[@Clyntax]

@@PinkeySuavo No, 0.5^(1/3) is already the solution to be written down and should not be resolved further unless you want to spend the rest of your life writing down digits.

[@PinkeySuavo]

@@Clyntax yes but the initial question was to solve it "in head" and this solution doesnt allow it, but if we talk about maths solution then it looks really nice

[@Bleaksigilkeep]

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

[@GravellordNito]

As someone who used to work daily with separating funnels, 80% immediately felt like the right answer. At least visually

[@henk-ottolimburg7947]

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

[@peppolone]

Exactly, 50% of the volume corresponds to the cube root of 0.5, which is approximately 0.7937, or 79.37%

[@lumenart7328]

Before watching the video and going by the thumbnail i think its the third glass to the right. Reason being that if you imagine the cup becoming cylindrical the water line will rest precisely in the middle. I picked the 80% option and got it right 🎉

[@suntzupup]

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.

[@justdilka]

I'm a martini glass half full kinda person

[@Minetendo_Fan]

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

[@Matty0311MMS]

Maybe @justdilka is in a quantum superposition of both?

[@chobies5383]

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

[@meki___6881]

Probably just a 12.5% full Person

[@rorywquin]

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]

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.

[@mattjanzen2344]

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.

[@Dexaan]

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.

[@andrewguthrie2]

We have the same problem with pubs serving short measures in straight beer glasses. These don't go to a point at the bottom, but are wider at the top than the base. When they draw the pint it will have an inch of foam head on it, and they'll say it's close to the top and can't get any more liquid in, but you're actually 10-15% short on a pint that costs upwards of £6 now. Its a rip-off.

[@petersmythe6462]

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]

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

[@nilton61]

A cone implies straight sides

[@katrinabryce]

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.

[@Lost_City007]

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

[@ThePowerfulOne07]

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

[@Temporary_yesyes]

it only goes to 80% to preserve battery life overall

[@ThePowerfulOne07]

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

[@amanavinash-fb4zk]

this video explains the opposite

[@chriswebster24]

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]

Rofl

[@rvsingh56]

I remember seeing a video somewhere, discussing the volume of a cone. I am also a math student so I instantly guessed it was 80% without calculation, as the lower part would contain lesser volume of the drink. Most of the volume of the glass lies in the frustum of the cone.

[@bootlegharold6696]

Presh that was needlessly overcomplicated lol

[@bscutajar]

6:00 I don't know why we are calculating the volume of the cone, the problem can be solved by considering the scaling of volume with the linear dimension

[@ShotgunLlama]

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]

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]

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

[@protoman1365]

@@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]

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

[@castleanthrax1833]

I've never seen a Martini glass as anything but conical in shape.

[@victoriamacarthur8906]

YESS I was right, my reasoning was with a 1d glass it would be at 1/2 mark or 50%, 2d would be 2/3 or 66%, go 3d would be 3/4 or 75%, round that up to 80 and you get the right asnwer. I’m not even sure at all if that was true but it worked.

[@hglundahl]

1:07 My answer 70 % presupposes basically a 45° angle, wasn't aware Martini glasses came in very different angles (I drink beer more often than Martini).

[@MiccaPhone]

the angle is irrelevant, 70% is always wrong, 80% always right - for all angles.

[@hglundahl]

@@MiccaPhone Thank you!

[@shruggzdastr8-facedclown]

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

[@JLvatron]

Math for Alcoholics! lol!

[@jensraab2902]

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]

iDz tHe BezZt kInDuh mAtHhHhHhs

[@JLvatron]

TeeHee-ous replies!

[@GAIS414]

I'm from Sweden, at dinner parties we traditionally drink vodka (Brännvin) from conical glasses. I learnt this trick many years ago from a teacher who said he always asks for half a glass every time the host came to refill. That way he could remain relatively sober but still participate in the festivities.

[@Bugrick92]

The tricky part is that we see this as a 2D triangle but its actually a 3D cone

[@cdarklock]

This is exactly the kick in the pants I had as I was watching the solution.

[@amanavinash-fb4zk]

0:04 if you look closely it is actually a 3D figure

[@ryalloric1088]

Yeah, the intuition of 70% is actually really close to the answer in 2d (√(1/2)=~70.7%)

[@weevilinabox]

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.)

[@maxc300es]

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

[@annoyingbstard9407]

Did you really do that? 😂

[@MiccaPhone]

Exactly, this is what fools the intuition - including mine.

[@Lowkjustthatguy]

@@jwdorybro has NEVER heard about a picture

[@Poggersvale]

Is bro a flat earther

[@maxc300es]

@@annoyingbstard9407 yeah xd

[@RobbieHatley]

At 0:48, off the top of my head, I'd say about 80%, and not dependent on angle. But let's keep listening. Ok, a 3:09 I see my guess was right. So how did I get the right guess? It's a square integration thing, so volume is going to be proportional to cube of height, by the integration formula for powers. 0.7^3=0.343, 0.8^3=0.512, 0.9^3=0.729, so 80% is the closest. Though, it's actually closer to 79.37%.

[@kylejacobs1247]

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]

I fell into the same trap.

[@MrJoerT]

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

[@lidarman2]

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

[@c.jishnu378]

Underrated.

[@drenzine]

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

[@musanim]

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.

[@Queenside_Rook]

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%

[@rajehhuawei1588]

You could use a similitude. The two cones are similar to each other. The ratio of volume is the cube of the ratio of height(in 2d the ratio of area is the square of ratio 10:02 ). So( r1/r2) ^3=v1/v2. And we get ,for sure,same result.

[@flameofthephoenix8395]

3:09 To be fair intuition based on sight doesn't really care if you call it a cone when our intuition clearly tells us it's a triangle, if you showed pictures of real cone shaped cups filled to those percentages we might have better luck!

[@suhaasvemuri7980]

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]

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

[@suhaasvemuri7980]

@@khaitomretro lol that too

[@Dejiek0]

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]

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

[@Dejiek0]

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

[@jhgvvetyjj6589]

The cone glass shape extends in both dimensions laterally and upwards, so it expands in 3 dimensions. The volume is therefore proportional to the cube of the height. Representing on a logarithmic scale, 1÷2 the volume is analogous to an octave below, or -12 semitones. The cube root of that would be -4 semitones, a major third down, approximately 4:5 ratio (as opposed to -5 semitones which is about 3:4, or -3 semitones which is about 5:6). Since 4÷5 is 80%, and the other available answers are outside the margin of error (below 3÷4 or above 5÷6), the closest answer is 80%.

[@rogue5882]

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

[@oneeyejack2]

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

[@Chris-hf2sl]

In similar figures, any area is always proportional to the square of one of the linear dimensions and any volume is always proportional to the cube of one of the linear dimensions.

[@peppolone]

Assuming a cone with a 45-degree angle where the height and radius are equal, we can disregard π and 1/3. If a cone has a volume of 1000, 50% of the volume corresponds to the cube root of 500, which is approximately 7.937 on 10, or 79.37% of the height. This also applies to a cube.

[@dennisdesormier6886]

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.

[@Lord-Sméagol]

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

[@lefthanded3512]

Wow, surface area really makes a difference

[@drelijahmikail3916]

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.

[@YardworkWithJohn]

At 3:12, I instantly realized that I was estimating half full based on a prisim, not a cone.

[@annesolarlunar6142]

I just visually imagined the cone glass into cylinder glass and imagined the liquid to go down, hence 80%.

[@ozargaman6148]

2:45 Some people probably cheated but I assume that most of those who calculated (including me) did it after choosing, only as a way to verify the answer and maybe learn something from it.

[@user-pr6ed3ri2k]

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.

[@matthewboire6843]

0:58 I think it’s that one, I think I remember something about angles that mean this is it.

[@flameofthephoenix8395]

1:24 You know, I hadn't even considered that the slope might be irrelevant. Hm, I'll check. The formula is a=(PI*r^2*h)/3 I believe, so if r = h*s where s is slope then a=(PI*(h*s)^2*h)/3, because we're squaring h*s this can be simplified to the square of h multiplied by the square of s making it a=(PI*h^2*s^2*h)/3, this can be further simplified by turning h^2*h into h^3 making it a=(PI*s^2*h^3)/3, the multiplication by PI and division by 3 are irrelevant since they are constant no matter the value of s so ignoring them makes it a=s^2*h^3, next to find the percentage we'll just determine the height of the glass to be cbrt(100/s^2)=h and the height of the liquid should be cbrt(50/s^2)=h this means that in order to determine whether the percentage is consistent no matter the slope we just have to determine if cbrt(a*x)/cbrt(a*x/2)=cbrt(a)/cbrt(a/2), we can check this easily, first we'll simplify everything to cbrt(a)*cbrt(x)*cbrt(2)/(cbrt(a)*cbrt(x))=cbrt(a)*cbrt(2)/cbrt(a), we can now divide both sides by cbrt(a) to get cbrt(x)*cbrt(2)/(cbrt(a)*cbrt(x)) = cbrt(2)/cbrt(a), now we'll multiply everything by cbrt of a because I failed to notice that I could have just cancelled out the terms instead getting cbrt(x)*cbrt(2)/cbrt(x)=cbrt(2), now we can cancel out the cbrt(x) division and multiplication to get cbrt(2)=cbrt(2) which is a true statement meaning that slope doesn't matter.

[@Bavarianscience]

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%.

[@iluxa-4000]

Before watching further, here's my reasoning. Keep in mind - I am not consulting any books or the internet rn, just using whatever is in my head, so I might be wildly incorrect. So, let's start with the volume of a cone, since I don't remember the formula. What is a cone? It's a triangle rotated around the center of the circle. The triangle's are is 1/2*r*h. Since we rotate it around a circle, and the circumference is 2*pi*r, we get V=pi*h*r^2. Now for the tricky part. We can't just plug in half of the height and expect the result because r varies based on h. How exactly? In any crossection, r/h=tan(fi), where fi is the angle of the cone. Since the angle remains the same, then the ratio is also always the same. We are talking about the height, so let's express r through h, using the expression above. We get r=h*tan(fi). Plugging into the original expression, we get V=pi*h*(h*tan(fi))^2, or V=pi*h^3*tan(fi)^2. Since pi and tan(fi) are constants, for the ratio V1/V=0,5 to hold true (V1 is the amount of liquid), we can ignore them. So we get h1^3/h^3=0,5, where h1 is the height of the drink we're looking for. 0,8^3 is 0,512, which is roughly 50%, so my answer is 80%. At first I thought it would be 70, but we'll see. Ok, So my answer was correct! Although my formula for volume wasn't lol But still, not bad

[@FosukeLordOfError]

7:57 I need more math in my life I was so close to this before I watched this part. Getting the ratio of volumes was the part I missed because I was trying to solve a specific case instead of the general.

[@mordechaisnyder7808]

Whew! Glad to be in the 28% that selected 80%! When I saw the results leaning so heavily to 70%, I assumed I’d totally botched it. Great puzzle Presh!

[@billjohnson3858]

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).

[@frumbert]

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

[@r.markclayton4821]

[@feynthefallen]

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.

[@0dd701]

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%

[@nzeches]

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

[@frankhooper7871]

Yay! My guess without any calculation was correct. ETA: I've had a similar conversation with a friend about the comparative levels of coffee, milk and foam in a cappuccino. Edited again to point out that at 7:05 you referred to the radio of the radia - by my reckoning, the plural of radius should be either radii (that you did use at 8:25) or radiuses (which I abhor as a plural LOL(

[@hephaistion3]

One of your best videos! ❤

[@hippophile]

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...

[@johnrains8409]

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.

[@NateHays]

Single scaling factor common to all three dimensions, so volume goes as x^3. Want V = 2V(x^3) so x = cube root(1/2) = 0.7913.... I think it is helpful to see problems firsst in terms of scaling, translation, rotation, polar/cartesian, or any other transform that simplifies. Don't need to look at integrals, formulae, etc. Presh, it would be great if you evangelize on this approach. I've picked it up from how physicists think. I wonder if anyone knows a good reference on how to do it. Maybe the way Div, Grad, Curl and All That by Schey brings vector calculus within easy grasp.

[@frankjohannessen6383]

it depends on the angle...when it goes towards 0 degrees then your dealing with shape that goes towards a pipe (with radius going towards zero) which means the height for half of the volume goes towards 50% height. When the angles goes towards 90 degrees then the width and depth of the top of the glass go towards infinity, which means the height for half the volume goes towards 100%.

[@frankjohannessen6383]

not one of my brighter moments. I forgot to think of the fact that even though the radius at the top of the glass goes to zero/infinite, so does the ratio at the top of the liquid and at the same rate. So the ratio between the radii remains the same.

[@anonymous_4276]

You can also think of the partially filled glass as the original glass but shunk down in all 3 directions by a factor of x. So in essence it has struck down by a factor of x³.

[@umchoyka]

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.

[@alfredtrietsch215]

As already pointed out in several reactions, the calculations can be simplified knowing that the ratio of the partial volume to the cube of the height is fixed. This is the case when the bottom of the volume is a point and the top surface area of any partial volume is some constant times the square of the height. This provided, the shape of the top surface does not matter, so it might be square, triangular or any weird shape you might imagine. If the orientation and general appearance of this "horizontal cut" shape stay similar throughout the height, the "sides" of the volume can be described as a bundle of straight lines from the bottom to the top edge of the volume. E.g., simple square or triangular pyramids fall in this category.

[@fullfungo]

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.

[@KiranRajagopalanMusic]

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

[@N8570E]

Off the top of my head, I would say number 3 from the left. 80%. And angle doesn't matter. Thank you for your efforts. May you and yours stay well and prosper.

[@mrKreuzfeld]

This problem does not depend on the shape at all, any object scales by it s scaling factor raised to the number of dimmensions the object spans.

[@johnmorriss5308]

If you use "Height and Cone Angle" as the parameters to define the dimensions of the glass, and realize that volume scales as the cube of the linear dimension, then you don't need to do any trig or involve pi in the calculation. Just cube root of 0.500

[@mathewmunro3770]

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.

[@BlueHighlighter-ew8qd]

"Is the glass half-empty or half-full?" "Neither. Air, plus, you don't split a cone shape equally like that"

[@first_namelast_name4923]

I did the calculation, but not because I wanted to cheat, but I enjoy doing calculations - and this is the reason I like this channel. I did the calculation to see whether my guesstimate was right, before I un-pause your video, just like I usually do with your videos ;-)

[@StefanVeenstra]

Was on the fence between 70 and 80, though admittedly leaning towards 70. Here's another question: Ever noticed how often the correct answer in multiple choice problems tend to be option 2 and 3? There's probably a fallacy, yet I notice that pattern and recognize I take the other options with more grains of salt unless it's obvious they're correct.

[@NoLongerBreathedIn]

Also useful to know this is that a major third is ideally 5:4, and three of them make a really good approximation to an octave (2:1).

[@jensraab2902]

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!

[@XVYQ_EY]

It definitely depends on an angle, as smoe kind of ratio of H & R determinates the angle. 90° filled to 50% height is 25% volume, and 75% height is 58,(3)% volume. 60° angle gives us 50% volume at 66,(6)% of height.

[@joshuaanoruo973]

That's not true

[@MorgurEdits]

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%

[@MinhAIPet]

Fun fact: A 95% height glass would only be about 85,7% full, meaning you can pour in a 50% height glass and it won't spill.