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Imbibition Equations (October 2024 Challenge Question)

Posted September 30, 2024 12:00 AM
Pathfinder Tags: challenge question

On a table in front of you is a 750 ml decanter, roughly 1/3 full of old, rare and delicious whiskey. You wish to drink it.

Its owner says you can have the rest of the whiskey - that is, if you can accurately measure how much is left using nothing but a ruler.

You can pick up the decanter, but you cannot spill or pour it.

Can you find the remaining volume left in the decanter? How?

Answer (10.24.24)

Assume the decanter has straight sides and a non-concave bottom (a rarity, yes).
The area (A) of a square or circle is easily found with a ruler.

Measure the height (H) of the whiskey. The whiskey has a volume of AH1.

Invert the bottle and measure the height of the empty bottle space. This volume is AH2.

You don't even need to know the volume of the decanter - although in this case it is helpful to understand how far off your imprecise measurements might be.

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#1

Re: Imbibition Equations (October 2023 Challenge Question)

09/30/2024 12:33 AM

Yes, dip the ruler in the decanter, mark the liquid height...whip out your matching decanter, fill it to a matching height, and measure the contents with your handy lab beaker...

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#2

Re: Imbibition Equations (October 2023 Challenge Question)

09/30/2024 4:21 AM

You choose a very large (and perfectly clean) ruler, which has a density greater than the whiskey, and, has a rectangular cross section. You break it into pieces, and, use one of the pieces to measure the cross sectional dimensions (and the extension at the end if there is one).

You now start to put the pieces in the decanter in order until it is nearly overflowing; finally poke the next piece in 'til the whiskey reaches the brim; the level showing on the ruler at the top of the whiskey tells you how much of the ruler was needed to fill the decanter, and, simple arithmetic gives you your answer.

I'm hoping that the owner will accept the description of the method, without demanding an actual demonstration.

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#3

Re: Imbibition Equations (October 2024 Challenge Question)

09/30/2024 8:33 PM

Let's say we know the width, W, and thickness, T, of the ruler. If we insert the ruler into the whiskey, the top of the liquid is distance L from the bottom, so the ruler has displaced V = L x T x W volume.

If we mark the decanter before and after insertion, and the liquid rises distance D, then we know that the surface area A = V / D. Finally, if the height of the whiskey is H, then the total volume Vt = H x A.

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#5
In reply to #3

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 12:32 AM

That's only true if the decanter has vertical walls on all sides.

The one illustrated on the original challenge appears to be close to that, but it's clearly over 1/3 full, so does not accurately illustrate the challenge!

Our only decanter has a shape closer to that of an hourglass, so this definitely would not work.

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#4

Re: Imbibition Equations (October 2024 Challenge Question)

09/30/2024 11:14 PM

Measure the depth of the whiskey, say Y mm, and then the depth to the 750 ml line, say X ml. Record the readings in your mind. Put the stopper in the bottle and invert it. Measure the depth of the whiskey. You have already broken the rule by using your hands and brain. These your friend must forgive or the problem becomes impossible to solve. Assuming forgiveness realize that the cross sectional area is constant and can be ignored. The difference in the two readings is half the bottle or 375 ml. Then X mm equals 375 ml or 1 mm is 1/375 ml. Multiply this by Y to get the volume of the contents. Now excuse me while I enjoy my own cheap bottle of whiskey.

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#6

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 4:56 AM

After studying the problem closely,the easiest way is to get a heavy steel ruler,2 pounds or so,whack the owner over the head from behind,drink the whiskey and leave.He will wake up with a terrible headache and presume he drank the entire decanter.Call your next case..(of whiskey,that is.)

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#8
In reply to #6

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 5:24 AM

On a more serious note,put the cap back in the bottle.Invert it and measure the height when inverted.Then measure the height upright.This will tell you the volume of the empty part of the bottle after a few simple calculations.Add the two measurements and you will have the original volume of the bottle,presuming the bottle was totally full.Now by carefully measuring the area of the liquid in the bottle,using the meniscus of the liquid as the measuring point,and presuming the bottle is perfectly square inside,use this value to calculate the remaining volume.This is presuming the bottle has a flat bottom,not concave,and the stopper is solid,not hollow.

The difference in the levels will give the ratio of the remaining liquid to the original volume.

I may change my mind when I get off of this inverting table.

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#10
In reply to #8

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 9:08 PM

OOPS! Posted in error.

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#11
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Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 9:22 PM

On second thought,place the plugged bottle on a flat level surface in a level horizontal position.

Measure the level of the liquid from the level surface,and measure the empty space above the liquid.

If they are equal,then exactly half (325Ml) of the liquid is remaining.

If the empty space is 1/4 of the full space,then,3/4 is remaining,etc,etc.

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#12
In reply to #11

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 9:49 PM

This will not work with all decanter shapes.

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#15
In reply to #12

Re: Imbibition Equations (October 2024 Challenge Question)

10/02/2024 7:16 AM

If we are free to design the canister any way we please,then I would choose a different shape.

I would choose one with graduated markings on the bottle to make it simple.

I was going by the image presented by the OP.

The method I described in my previous post would be accurate at exactly the half way mark,if the container was symmetrical.

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#16
In reply to #8

Re: Imbibition Equations (October 2024 Challenge Question)

10/24/2024 11:24 AM

Now this! THIS is the answer we were hoping for.

Bravo.

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#7

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 5:19 AM

You take the decanter to the king (emperor, president: choose your local chief), i.e. ruler, and, ask him how much whiskey he thinks is left; woe betide anyone who disagrees with him.

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#9

Re: Imbibition Equations (October 2024 Challenge Question)

10/01/2024 5:57 PM

I have a few silly questions:

  • Can one write notes (pencil and paper) to make calculations or must only the proscribed ruler be used to measure the liquor?
  • Also how accurate of a measurement? Without using a ruler, I can say there is about 250 ml of liquor in the decanter. I also say that is accurate enough.
  • The remaining volume in the decanter is about 500 ml. (Don't you love ambiguous puzzles? The poser can choose any of several answers as correct.)
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#13

Re: Imbibition Equations (October 2024 Challenge Question)

10/02/2024 12:57 AM

A decanter of whiskey is never filled to the top, there is an unknown volume of space beneath the stopper...

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#14
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Re: Imbibition Equations (October 2024 Challenge Question)

10/02/2024 7:09 AM

Based on what you say,then the ratio of top of the square shape to the bottom of the canister will be 750 Ml, the total volume,that is if the bottom inside is a flat bottom and sides are square and smooth then the ratio of heights will give the answer.

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#17

Re: Imbibition Equations (October 2024 Challenge Question)

10/27/2024 11:25 AM

Another method just occurred to me:

On 3rd thought,that will not work either.Funny how ideas look different when you write them down and read them back to yourself.

Flaws in logic are more apparent that way.

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