Very few people can solve. Which martini glass is closest to 1/2 full?
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- Опубліковано 15 вер 2024
- Just how full is your cocktail glass anyway? The answer will surprise you.
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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.
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).
Pictures are not naturally 2d. They are artificially 2d to mislead and confuse.
@@StRanGerManY What's the shape of the screen you're viewing this on?
@@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
@@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.
I am half way through the video and forgot about the third dimension
Yep, me too.
That was my exact mistake
Same. 😅
Was thinking between 70% and 80%, but closer to 70%.
Yea I guessed that is what most people though. That is why I guessed 80% as it was a cone.
But first, we need to talk about parallel universes.
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.
another round, please
I was going to say the same thing! lol Real world experience is priceless!
@@spacewolfjr another martini, Paul?
What if a mathematician shows up there?
@@abhishankpaul 2^(-1/3)
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
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.
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!
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%
@@AiNaKa yea that's what I thought
I guessed 80% because it looked like that was the closest to half full. Not a bartender or anything.
Technically 100/2^(1/3)% = 79.37%
The angle doesn't matter cause stretching the glass horizontally and depth-wise will preserve the proportions.
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?
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.
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.
@@martianunlimitedbut both of you are doing calculations 😂
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.
@@stuchly1how much is 2+2?
Is it 4 or 6? You are not allowed to do any calculations.
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.
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.
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%.
Ah yes a 2D glass
I’ve never heard of a 2D liquid. The question implies that it is in 3D.
I'd love you to pour me a 2d drink! How many ml go into a 2d martini glass? 😂
@@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.
Cone is really good shape of glasses for bars’ owners😀
And cubes. Don't forget cubes...Ice cubes. Plenty, plenty ice cubes.
If they don't fill it to the top, it's a sham!
„Con” is literally in the cone name so
@@gregoryt1139martini on the rocks it is...
Without using a calculator I realized the answer was the cube root of one half. Then I needed a calculator.
I had it memorized from when I used to to teach physics
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.
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.
@@ThatFoxxoLeoI like that explanation. Thank you.
512 is 8^3, so .512 is .8^3, so .8 is close to cbrt(.5)
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%.
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.
That's the way I did it
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).
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.
I'm even more impressed by the fact that the 50% height is only 12.5% full.
All the glasses are 100% full. It's just a matter of what they're full of.
my glass is filled with 90% water and 10% bose-einstein condensate
@@user-pr6ed3ri2k Mine has 10% quantum foam
@@fatmccat1513 what's the other 90%
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 lol this is what we thought when seperating a cone into 3 equal pieces
*Remember that the real-life counterpart of that diagram is 3-dimensional...*
Yeah, that was what got me, lol. 2d diagrams are a nasty trick.
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.
That explains why my iPhone thinks 80% battery level is considered full!!!
it only goes to 80% to preserve battery life overall
@@Temporary_yesyes trust I know and for full performance and capacity as well!
this video explains the opposite
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%.
Rofl
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
I'm a martini glass half full kinda person
So are you 1/8 full or 80% full?`
Maybe @justdilka is in a quantum superposition of both?
I'm a " Martini to the half way point"
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.
Can work tho with 2d drawing animation I guess
@@mofprailes7140 The question was clear. It said LIQUID.
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
In 2 dimensions I can color the surface of the glass with a marker.... That is liquid.
But hey 😂 I am joking here 😉
My guess was based on 2d as well... i just saw it as a triangle
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.
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
People that clicked 70 percent because they thought of the martini as 2d and not 3d here 👇👇👇👇👇👇
Did you really do that? 😂
I never heard of a 2d martini. What are you a flat lander?😅
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.
The angle of the taper DOES NOT MATTER as long as the sides are straight.
A cone implies straight sides
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.
Now hol up. Who said it has to be a cone? What if we're looking at a cross-section of a triangular prism?
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.
Doesn't make a difference. Any prism is 1/3 cubed route of height.
A cone is just a many-sided prism.
AD0TJ
@@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?
@@protoman1365 Ah my b, mixed up prism and pyramid
I've never seen a Martini glass as anything but conical in shape.
INtuitively I said 80%, since the volume goes up the wider the glass. Nice to see it explained.
Presh that was needlessly overcomplicated lol
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.
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
Math for Alcoholics!
lol!
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!
iDz tHe BezZt kInDuh mAtHhHhHhs
TeeHee-ous replies!
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.
Pretty easy question if you've done some cooking and have used a Tala measuring cone for ingredients.
@@khaitomretro lol that too
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.
Well yeah but the question specifically said to not calculate it for this reason.
@@meta02 Correct. I guessed 80% before going on with the calculations.
The tricky part is that we see this as a 2D triangle but its actually a 3D cone
This is exactly the kick in the pants I had as I was watching the solution.
0:04 if you look closely it is actually a 3D figure
Yeah, the intuition of 70% is actually really close to the answer in 2d (√(1/2)=~70.7%)
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.
The misleading part of this question is that it is presented as a 2D problem visually, when in fact it is a 3D problem.
I fell into the same trap.
In 2d 80% is still the closest answer, right?
@@MrJoerT I think it is 70.7% for 2d case. Solve the 2d version, 1/2 = x^2.
Underrated.
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
Perhaps more alarming is that when you sip that 20% off the top you are half done...
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.
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.
Without having to rearrange, the volume scale factor for a 3d object is (the length scale factor) ^3
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
i thought for just 3 second and thinks the answer is 80%. i imagined pouring
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!
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
A perfect visual example of integration and differentiation. The outer lines are linear, the circle is squared and the volume is cubic. 1/3 x² is the derivative of x³.
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
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.
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...
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.
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.
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!
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.
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%
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.
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%.
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.
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.)
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.
Height is proportional to the radius, so volume is proportional to the cube of height. Hence 1/2=v2/v1 = (h2/h1)^3
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
I got shown this back when I did bar work years ago. Take a one-measure and pour into a double-shot conical glass. Now ask everyone if ANOTHER one-measure will fit or not? It LOOKS like it won't, but the names give it away. Two one-measures will always fit in a double-shot glass.
The volume of a cone or pyramid is h x b / 3, with the height linearly proportional to fullness and the base b = pi r^2 quadratically proportional. So the volume is proportional to fullness cubed, cube root of 1/2 is 0.7937005259841, and the answer is "closest to 80%". Cross check: 0.8^3 = 0.512.
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.
My guess was the right answer. I did not CALCULATE it, but I very much do have thought thing I created for myself that I put geometry related stuff through, and it's essentially just a shout of "REMEMBER THE SQUARE-CUBE LAW, TAKE IT INTO ACCOUNT". So my brain had an initial guess that the shouty rule caused me to adjust upwards by an amount that seemed reasonable and that got me to the right spot.
I had a similar experience. I initially said 70% from what my intuition was telling me, but then realised it would be 3D, and thus there would be a greater disparity in the difference between height and volume. If I were to pick any number I probably would have said 85% and been wrong, but given 80% and 90% the answer was obvious.
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.)
Well why did you assume the martini had 10 days to fill instead of 5?
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
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).
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
just from the thumbnail we pour the 80% into the 60% until its 100% and then the 70% into 90% until its 100% then u add up the difference in height of both emptied glasses to get the result height
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
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.
Area of booze for any horizontal slice is gonna be x^2 compared to height. If you do the integral of that, you get x^3. You're looking for x^3 = 0.5, and thus cuberoot(0.5), and thus 1 / cuberoot(2). Cuberoot(2) is about 1.25 (closer even to 1.26) (which is also why a major 3rd in music has a frequency ratio of about 5/4). 1 / 1.25 is 0.8. So 80%.
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 ;-)
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.
That's not true
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.
Before I watch it: volume is some fraction times height times square of a diameter, changing height we are getting similar triangles, so volume is just proportional to third power of height. 8x8x8 is 2^9 is 512, so 0.512 is close enough to 0.5 that I guess it is the right answer
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(
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
I answered this intuitively, and calculated it later by cubing the decimal values of the percentages.
I picked 80%.
Cuemath is doing a wonderful job, thanks for calling it out 💪
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.
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.
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.
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%.
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.
Excellent problem and solution.
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.
Radius a linear function of height. Surface area proportional to square of radius so square of height.
Integrate over height to get proportional to height cubed.
Or just - everything is linear and volume is a cubic so must be height cubed.
So volume of glass (say 10 high) proportional to 1000, height = 8 proportional to 512 so about half.
There is a bar trick where you compare the circumference of a glass with a stack of objects next to it. People find it very difficult to judge the length of a circumference. Make an 'OK' sign with your thumb and finger. Now stretch your fingers out as straight as you can. The length of your two fingers looks so much bigger than you would have estimated from looking at the circle.
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).
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.
My intuition based on initial feeling would be around 70% of the height, even with the variable angle option. I guessed 80% because it looked more unlikely and would make a more interesting video though.
I also have one of those coneshaped measuring cups that I use quite often, and while it starts with a smaller flat base, it goes quite high up for the halfway mark.
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!
Had a head start thanks to Numberphile's video yesterday on the exact same problem. One could construct an Euler diagram of the audience/viewers/subscribers of this channel and that one
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
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.
your solution kind of confused me, so I am going to write how I solved it. V1 = volume of below cone, x, r - height and radius of the below cone, V = volume of cone, h, R - height and radius of the cone:
V1 = r ^ 2 * x * Pi /3. From the similarity of the triangular cross-sections we get: x/h = r/R. Multiplying by h we get: => r = x * R / h.
Replacing this value of r to the first equation we get: V1 = x^3 * R^ 2 * Pi / (3 * h^2). V = R^2 * h * Pi /3.
Since we need 50 %, we can write: 2 * V1 = V. So 2 * x^3 * R^ 2 * Pi / (3 * h^2). V = R^2 * h * Pi /3.
When terms cancel out, we get: 2 * x ^3 / h^2 = h. Dividing both sides by h/2 we get: x^3/h^3 = 1/2. so (x/h)^3 = 1/2.
x is p (%) * h so we get: p * h / h = root 3 of (1/2). h cancels out and we get that: p = root 3 of 1/2. which is equal to 79.4%
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.
I only guessed 80% because my wife and I did a test to measure our jiggers in ounces for cocktails and it was crazy how much more volume a full jigger was to a close to full one. Jiggers weren’t cones but led my intuition for this problem.
I think you could have given the following answer options {66.67%, Answer cannot be determined because it depends on the base angle of the glass, 75%, 80%}.
Divide the shape into smaller triangles and count how many are colour yellow and how many in black. A quick comparison and I would say it's between 70% and 80%...nearest to 70%