How many candies are in a jar - packing efficiency and volume?

Factual Questions
1

I’ve been trying to come with some mathematical formula to determine how many candies ARE in the jar. I’ve came up with one but there are two problems with it and I was wondering if you guys could help me out with those:

Here’s my formula:

Let x represent the total volume the candies altogether occupy in the jar

Let y represent the total volume of air spaces between candies in the jar

Let v represent the total volume of the jar

Let d represent the density of the candies

Let m represent the mass of the jar (glass not included)

So from here I created an algebraic system:

x+y=v (The volume of the candies plus the volume of air in the jar is going to equal the volume of the jar)

dx+0.00128y=m (The mass of the candies inside can be determined by density times volume. The mass of air inside can be determined the same way except the density of air is 0.00128 g/mL. There fore the mass of air inside is 0.00128y. So the mass of the candies inside plus the mass of the air inside will equal the mass of the jar (excluding the glass).

Now d, v, and m must be predetermined.

I’m actually going to the lab to find out the density for all the different candies.

The problem is with v and m. Because v is going to have to be known and there are so many types of jars that I don’t know how I would determine the volume of those jars.

m is going to be calculated by knowing the packing efficiency, how much of the volume is actually occupied by candies and not air. Once I know the packing efficiency, I can average out the densities. Suppose the packing efficiency is 70% (meaning only 70% of the jar’s volume is occupied by candies, and 30% is air). You can do 0.7d+0.3(0.00128). Suppose I know the density of the candies (which I will) and it is 1.5 g/mL. I can do 0.7(1.5)+0.3(0.00128)=1.050384. So the average density inside the jar is 1.050384. Now we know that volume times density equals mass. So if we know what v equals, than we can multiply that by 1.050384 and find the mass. So that would give me m.

d is easy to figure out. I’m going to go to the lab and measure every candy (I have all the candies).

But I don’t know how to figure out v. How should I go about calculating the volume of the jar because jars are irregular objects and I’m not going to be able to use water displacement for that.

And for m, I’m going to need to know the packing efficiency. So if I know what the packing efficiency and volume is, then I it’s easy from there. But that’s the problem. How do I figure those two things out?

Once this system works, I can just divide x by the volume of one candy which I will have pre-measured anyways. So the purpose is to find x.

By the way I’m a freshman in high school taking math classes accelerated by two years (Geometry B/Algebra 2B).

2

Why can’t you fill up the jar with water then dump it in a measuring cup to get the volume?

3

Because I won’t be able to do that in real contests.

See I can find the density out easily and if anyone knows about packing efficiency that can be solved too.

However, the volume of the jar can’t be measured by water displacement at all. Think about it, no store will let you do that. Most won’t even allow you to measure but okay, if there is a way to do it by measuring with just a ruler, that could work.

4

Measure or estimate the height of the inside of the jar. Visualize a cylindrical jar with about the same volume as the real jar. Here you’re trying to make two smaller quantities equal: the amount inside the real jar but outside the cylindrical one, and the volume outside the real jar but inside the cylindrical one. Those are smaller quantities, so any errors will tend to be small.

Now estimate the diameter of the cylindrical jar, and use that to get your volume estimate.

5

Thanks for the input. I don’t understand the whole smaller quantities part.

So let’s say I visualize a cylindrical jar with about the same volume as the real jar, what do I do after that?

I’m sorry, it’s just hard to understand if you could please clarify what you said, that would be great.

Also, do you know how to go about packing efficiency and if there is a formula for that?

By the way, what do you think of my formula that I have so far? Is it good?

Sorry lots of questions.

6

Estimate the diameter of that cylinder. Then the volume is diameter * diameter * height * pi / 4. I’m going to dinner, so i don’t have time to go over the rest of your post, sorry.

7

No it’s fine. It’s just that won’t that give me an inaccurate answer because the jar is not cylindrical. I see what you are saying, but first of all it is not easy to just visualize a cylinder that has a the same volume because the jar is not really a cylinder. Second that will be too inaccurate. Maybe I am just missing the point. Can any other user help me out here?

8

So you should clarify with the people running the contest: Just what are the rules? Is there a rule that you may only use methods that a supermarket manager would allow, if you were buying a jar off his shelf?

So, I get that maybe you are presented with a full jar and you aren’t allowed to dump it out to fill it with water. But could you seal off the top carefully with tape and then immerse it in a tub of water, to measure the runoff? IIRC a guy in Syracuse tried something like that, successfully, some years ago. The questions here is, what are the rules you have to follow?

Note that this method may require you to run through the streets naked while attracting attention to yourself by shouting stuff in Greek at the top of your lungs. You would need to decide if you want to do that.

BTW, it’s very different In Real Life. You see those contests sometimes, How many candies (or marbles) are in the jar? What they don’t tell you is, those jars usually have a larger ball (a ping-pong ball or tennis ball) hidden inside of them.

9

[tangential anecdote]This reminds me of a test we had in Chem 1A lab once. It had no right answer that I could ever see, so I guess the teacher was just looking to see how students tried to deal with it. Given a tub of dry ice, we were to take a chip of that, and determine its density. We had learned, of course, techniques like immersing the item in water in a graduated cylinder to get its volume and weighing on a scale to get its weight. But just try techniques like that with a piece of dry ice!
[/tangential anecdote]

As for the problem of the OP: What you seem to be saying is, you simply don’t have sufficient data to solve the problem – depending on just how much you do or don’t know about the shape of that jar.

10

Using calculus find the volume of the jar. Buy the exact candies and put them in a smaller container. Find the volume of the container. Count the candies in the container. For example, say the volume of the jar is 100x, and the volume of the container is 5x. Multiply the number of candies in the container by 20 and you have won. It worked for me and I won. The jar was shaped like a large light bulb, therefore it was necessary to use calculus to find the volume. Good luck!

11

What shapes are the candies in? If they are round, then the theoretical maximum packing efficiency is a little more than 74%. Check the wikipedia article on close packing.

12

harmonicamoon, how will I know that the actual jar is 20 times bigger than the container. Or what I mean is, how am I supposed to know how many times the actual jar is larger by than the smaller jar.

And Senegoid, this formula is designed to work so that you can just go anywhere where the most you can do is measure parts of the container. I will be able to find the density and volume of the candies as I am planning a day in the lab for that. By the way, I’m not actually in this situation. What I’m doing is preparing for this kind of situation so that when it comes, I can just plug numbers into the formula I have created. I think it’s possible.

But the biggest problem is the volume of the container. Even measuring won’t be sufficient enough because what if the object is an irregular shape? And honestly I really don’t want to have to measure because it’s not accurate and most contest holders won’t allow it. Aren’t most jars a set standard volume so you can just tell if it’s 8, 16, 32, 64, or 128 ounces? But then again some aren’t and it’s hard to tell.

I feel like the earlier poster was on to something about imagining a cylinder that will equalize two quantities, but the problem with that is the inaccuracy. How do I actually make that work. I think he has a good point. It’s just hard to do though.

Let’s not worry about the packing efficiency right now, or the ratio of space the candies actually occupy to the air space left in between candies, because that’s not a very critical factor in this system. The reason I say this is because most I believe are going to be around 60 or 70% anyways and I do know them for the most common candies anyways. For m&m’s and skittles I believe it’s 66.5% and for spherical candies like gumballs, it’s 64%. That’s what howcast says anyways. But even if they are using different candies, you can’t go to wrong by just estimating 60 or 70% since that doesn’t even affect the answer all that much.

The most critical part of this is knowing the volume of the jar because if the volume is off, then the answer is off since that’s what determines a lot of the story. Sure some contest holders will tell you the volume because they think that you still won’t be able to do it but many do not even allow you to TOUCH the jar. Now that’s going to be tough. But if someone can manage to tell me how to get a volume that’s at least close (like within 10 or 15 mL), I fine with that. I’m not counting on too much accuracy anyways. I mean my goal is for my formula to usually be within 20 candies if there are thousands of candies inside, and 5-10 if there are only around 100 in there so a few mL off won’t kill me.

13

I think we’re still unclear on what data you will be given, or what data you will be allowed to find out for yourself (by empirical means, if necessary).

You seem to be saying that:
(a) Your are trying to work this out in advance, but you know little or nothing about what the volume will be.
(b) You don’t even know what the shape of the jar will be.
© You believe that you will not be allowed to make any kind of measurements on the jar.

So the problem seems to be strictly an algebraic problem, not an experimental problem.

Can you at least expect that the jar will have some kind of label on it, giving its volume (in ounces, mL, bushels, pecks or whatever)? Or that you will at least be told what the volume is? (Or does that make the whole problem tooooooo easy?) If you have to figure the volume yourself, what raw data do you think you will be given from which to compute that?

Since you are a high-school frosh, but taking advanced math classes, we should also ask: Just what level of advanced math do you know? Do you know basic calculus? (Seeing as how harmonicamoon, above, suggested that you could use calculus.) Would you know how to do that? (Your use of the “dx” in the OP suggested that, at first, but on second reading, I don’t think you meant the calculus kind of dx.)

Do you have any advance knowledge at all on the shape of the jar? Do you know if it is a volume of revolution? (That means, if you take cross sections by slicing through the jar with a horizontal plane parallel to the table, will all such cross sections be circular?) Or might the jar have a square, or oval, or irregular shaped cross sections when sliced this way?

If the jar IS a volume of revolution, AND you know the jar’s profile (the shape of the cross section you would get by slicing with a VERTICAL plane that goes through the jar’s vertical mid-line) – that is, you know an x-y equation for that curve and it’s not too complicated – then you could find the volume by integration. That’s a first-semester calculus thing, if you know how to do that. (And if you’re allowed to use calculus in this contest.) If you’re allowed to, you know, actually get your hands on and measure the jar’s circumference at many horizontal cross sections, you could do this too.

14

The Computer Science version of this is “Teach a Martian how to smoke a cigarette.”

The gag is, you can’t because you don’t know what a Martian already knows. For every single possible instruction you come up with, the Martian can misinterpret it by not having some crucial piece of knowledge that instruction depends on. But since you don’t know what the Martian does know, you can’t tailor your instructions.

The lesson is Get your assumptions out in the open first, and then design.

15

[tangential sidetrack, or should I call this a normal sidetrack?]A comment on the side: If you are an advanced math type, and you really like this subject and are turned on by mathematical challenges, .AND. if you don’t know any Calculus yet: Get thee a Calculus textbook and start diving in! Beyond High-School-Level Algebra and Geometry, this takes Math to a whole new level, and a whole new pack of challenge. According to the extent of your math-geek buddha nature, you might find it to be a major turn on. Start deriving today! :wink:

(You don’t really need Trigonometry to get started in Calculus, but at some point in First Semester Calculus you will. Still, you can get quite a ways into Calc without Trig if you just skip the Trig problems. [/tangential sidetrack, or should I call this a normal sidetrack?]

16

This is mostly an algebraic problem. That “dx” was d times x. The density of the candy times the volume of all the candies in the jar. People my age are supposed to be taking Algebra 1 right now, but I’m taking Geometry B/Algebra 2B, and next year I’m taking Pre-Calculus. Geometry B/Algebra 2B is accelerated by two years. Last year I took Algebra 2A/Geometry A, and two years ago I took Algebra 1. Being told the volume would obviously make the problem too easy and most contest holders wouldn’t give you this information anyways.

But I was wondering if you guys could teach me a little calculus. You’re saying there is a way to figure it out using calculus. If there is, then go on and show me. I haven’t started any calculus yet but I’m very strong in mathematics. In fact, I’m trying to get into MIT. This problem will use lots of Algebra, some Calculus, and a little Chemistry (the whole mass, volume, density, and packing efficiency thing). Suppose the jar looks like this: http://www.google.com/imgres?start=69&um=1&hl=en&biw=1527&bih=840&addh=140&tbm=isch&tbnid=PS57ezeSI6lTbM:&imgrefurl=http://www.bottlebooks.com/questions/march2000/march_questions.htm&docid=zIa4vNHEv6UbWM&imgurl=http://www.bottlebooks.com/questions/march2000/March_23.jpg&w=226&h=322&ei=JydyUNCkIsPA0QH4lIDoDw&zoom=1&iact=hc&vpx=431&vpy=281&dur=954&hovh=258&hovw=180&tx=119&ty=146&sig=116961775635133810923&page=3&tbnh=159&tbnw=113&ndsp=36&ved=1t:429,r:15,s:69,i:53

17

Use alcohol (ethanol, methanol, shouldn’t matter) in your graduated cylinder, and prechill it in a bath of dry ice and alcohol. Nothing that someone with high school chem couldn’t think of in principle (in practice, maybe not so much).

18

This seems especially apropos to the problem of Documentation, especially the writing of end-user-level instructions: That no matter what you write, the dumb end user will get it all wrong. (Also seems to happen in the passage of knowledge from the programmers to the Customer Service guys too.) This explains the solution to documentation difficulties that has become so widely accepted these days: Just don’t bother documenting anything because it’s a wasted effort anyway. Or if you do document, just make it quick and dirty and perfunctory, because it will be misunderstood anyway, assuming anyone reads it at all.

When all else fails, DON’T bother reading the manual, because it will probably be less helpful than anything else you’ve already tried anyway!

:smack:

Oh, and for Anonymous User, the OP: Personally, I think the art of writing textbooks has largely gone to shit too over the past 30 years or so, and I’ve definitely seen it in College level Algebra books among others. If you decide to try your hand at Calculus, see to it that you find a GOOD text book. IMHO, that generally means an OLDER book, such as you might find in a good Used Book store.

19

According to Richard Feynman, the reason for shitty textbooks is rampant corruption–and that was nearly 50 years ago. I doubt anything has changed.

20

Well, I never thought of that. And I can damn well assure you that nobody else did either. It was kind of hilarious watching other people in the class trying various ways to make various measures of the chip of dry ice.

First of all, we all got rather tiny chips. Right there, anyone would be lucky to get even one significant digit of result. (Assuming anyone in the class even had the concept of what a “significant digit” was.) We had a super-sensitive scale that could weigh thing to the 0.001 g (or was it 0.0001 g?), but if you put the chip of dry ice on it, it would be evaporating fast enough (at that level of measurement) that you never got a stable reading.

Some students tried immersing it in a water-filled tube inverted into a tub of water. I think they were trying to measure the volume of evaporated CO[sub]2[/sub] that collected in the top of the tube. :confused: What happened, of course, is the the evaporated CO[sub]2[/sub] immediately blew ALL the water out of the tube.

The next day, I asked the Lecture Prof about it (a different person than the Lab Instructor). I swear to Og, he didn’t know that “density” was an attribute of the physical state rather than the chemical nature. He tried to give me the analogy that water is still water, whether in solid, liquid, or gas state, so the question really dealt simply with properties of water – or in this case, with properties of CO[sub]2[/sub]. :smack: and :rolleyes:

Oh, and as for Anonymous User: (Remember Anonymous User, the OP?): I think someone very recently posted some links of “Beginning Calculus Made Simple” type of web sites, IIRC. Might have been in that .999… thread. I’ll see if I can find them and post them again here.