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

Factual Questions
21

I’m well aware of Feynman’s screed on that from Surely You’re Joking, Mr. Feynman and I believe it. But I ALSO believe in the view that it’s all just been getting increasingly dumbed-down over the years. I saw it also in Survey-level History books, like we used in college Freshman American History classes at the JC. I went to the nearby Uni library (Cal Poly in my case) and found a REAL (older) textbook written by some of the Big Name History Professors, which I read in parallel with the assigned class text. My JC history teachers were Impressed As All Get-Out by this.

If Anonymous User feels the same about his Math texts, he should try the same. I suggested Used Book stores as a good source for older, better math texts.

One word of warning though: Older books tended to rely a little more on rote memorization of rules, and a little less on teaching to “Understand What You’re Doing RATHER Than Getting The Right Answer” (as Tom Lehrer put it). Their explanations were often more terse, and didn’t say much about Set Theory or Equivalence Classes or other such things, that the New Math folks seem to think should be introduced years before the student will actually use those concepts.

I have a College Algebra text by Britton and Snively published in circa 1942 (about ten years older than I am). It has an excellent chapter on Inequalities the like of which I’ve never seen since. After studying that, I was much better prepared to understand epsilon-delta proofs. It had a whole chapter on dealing with approximate numbers, ditto. Other topics were covered in greater depth than you ever see these days in college-freshman Algebra.

I had a Trig textbook from 1914. It gave reduction formulas for ALL SIX of the common trig functions. When I mentioned that to my Calculus II prof, she had never heard of reduction formulas for cot, sec, and csc.

22

Note to Anonymous User: Re: You asked if we could help you get started on Calculus.

Well, nobody here is going to write a whole beginners textbook for you (there are plenty out there) but we can probably give you various helpful tips if you want.

Here’s my first helpful tip, that I think should be just right for your current level: Have you learned much about functions? Typically (as I learned it in high school), these are covered a little too-superficially in high-school algebra, but much more thoroughly in Trig.

You really need to get a SOLID proficiency in the concepts and workings of functions – This is one of the core topics studied in Calculus. If you already have this, then be sure you get well-practiced in working with functions at the level you have learned. Pay attention of inverses of functions too.

Second, to be more particular: Learn about COMPOSITION OF FUNCTIONS – that is, if f(x) is one function and g(x) is another function, get practice in dealing with f(g(x)) and g(f(x)). When I learned Algebra, we touched on these but didn’t really get a whole lot of practice. When I took Calc I, we started with an introductory chapter that was a review of functions – everything we should have already known about them. Pretty much EVERYBODY in the class really bombed out in the section on Composition, including me. But as you get into Calc, you will be using composition A REAL WHOLE LOT. So practice that.

There. There’s your Beginning Calculus Lesson One.

ETA: For composition, you will also see notations like f o g (x) or g o f (x) (where that “o” is a little circle symbol, not the actual letter “oh” o).

23

Or maybe some kind of solid, prechilled, with a small enough grain size, something like sand/silica ?

24

President Obama, in debate with ex-Governor Mitt Romney, Oct. 3, 2012, University of Denver, Denver:

[A] teacher that I met in Las Vegas, wonderful young lady, who describes to me — she’s got 42 kids in her class.

The first two weeks, she’s got them — some of them sitting on the floor until finally they get reassigned. They’re using textbooks that are 10 years old. That is not a recipe for growth; that’s not how America was built.
Hi Anonymous User, welcome to SD.

25

I don’t think this has been answered yet. I just posted a First Lesson In Calculus (see above post). Summary: Get really good in working with functions, especially including composition of functions (which is usually covered only lightly in Algebra, at least as I learned it).

As for that link to all the bottle pictures: Well, there are a whole lot of bottles pictured there will all kinds of shapes. Some of them are volumes of revolution (as I described earlier), and some of them definitely are not. The most complicated volume of revolution I saw there is that bottle with the faces molded into the front of it (ignoring the trivial irregularities caused by those faces). On the other hand, the rectangular bottles are NOT volumes of revolution.

You could still use calculus to find the volumes of ALL of those, if you had enough raw data to nail down the actual shapes of the bottle. The technique involves integration to find volumes of solids of known cross sections (which includes volumes of revolution). This would normally come up maybe two to three months into a Calc I class. So I don’t think a detailed description would mean much to you at this stage.

I can give you a superficial description of the process, to give you a picture of what it’s all about. Suppose you lay the bottle on its side and slice it into thin slices, like you were slicing a loaf of bread. If you could find the volume of each slice and add those up, you have the volume of the whole. But you can’t.

So suppose you could find the approximate volume of each slice, and add those up. Then you would have approximately the volume of the whole bottle. If the bottle had some rather irregular sort of shape, this is what you would have to do. And if you made your slices thinner and thinner (and thus, more and more of them), you could get closer approximations because there would be less irregularity around the edges of each slice, the thinner the slices are. Taken to the extreme, you could make infinitely many, infinitely thin slices. They would then have “infinitely little” slop around the edges, and you would therefore get the exact volume. In calculus, you learn how, in effect, to do this mathematically, and you derive formulas for the volume of various kinds of shapes. Also, formulas for the areas of shapes with curved outlines. You need a couple of months of calculus before you begin to get into these techniques.

26

Leo Bloom, quoting President Obama in the debate:

I’m not sure just what vintage of textbooks had what quality, but I’ll stick with my thesis that the older textbooks tended to cover material in more depth – especially prior to the “New Math” stuff. To be sure, “New Math” put some emphasis on some specific topics that, today, elementary math students are expected to know (like Set Theory), which the older books didn’t cover at that level.

42 students, some sitting on the floor: Probably not such a good plan. Too little time to give students’ much individual attention or answer their questions. And Og help the poor teacher who has to correct that many kids’ homeworks!

27

Well Senegoid, thanks for your input. I will start learning some calculus and get comfortable with functions. Well, I’m going to bed now so I will see you tomorrow on this thread. Thank you!

28

I was just thinking the other day about, Why is it that what I know of Richard Feynman is only from links in posts or youtube? .

Now it all makes sense. :mad:

29

I was in 1st grade in 1961 and we had the first generation of texts with set theory in them. What a colossal waste of time. The first three weeks of every year we spent learning unions and intersections and then never used them again. You don’t need to learn that “0 = the null set” to learn to count. Rant over.

30

[Continuing fumster’s rant]

Hijack though this may be, it’s already come up in this thread, so I’ll run with fumster on this. This is right. Set Theory isn’t particularly useful until much more advanced math than this, at which time it becomes quite useful. Until then, fumster is right – you spend too much time studying it, too early on, with no particular use for it, and with no clear understanding of what it’s all good for.

I took Algebra I in 9th grade (circa 1965), where the first entire chapter was devoted to this, right at the beginning of the class. Thereafter, pretty much the only thing we did with sets was use them to write the “solution sets” for the equations we solved. For example, if x[sup]2[/sup] - 8x + 15 = 0, no longer are we allowed to write: Solution: x = 5, x = 3. (This is how solutions were written in that old Britton & Snively algebra text I mentioned above.)

Oh, no! Now we must write: x ∈ { 3, 5 } for no particular reason that would be clear to a beginning Algebra student. It wasn’t until some years later that I came to appreciate the usefulness of Set Theory, and I think that would have been a better time to study it. A brief mention of it an an earlier stage might be fine, but not to study it for three weeks just to write answers as solution sets.

Teaching three weeks of set theory in lower elementary grades? Gimme a break. Do they still teach the Five Fundamental Laws of Arithmetic? I’ve heard students are coming up without knowing that, and THAT’S a real shame.

I’ve been away from this for many years, obviously. Anonymous User, is that how they’re still teaching it these days?

[/Continuing fumster’s rant]

31

That part I bolded is the problem, not set theory. Good education builds on itself (and occasionally tears parts of itself down, when simplifications get revealed as lies and replaced by more correct models).

32

BTW, FWIW, here’s a good example of a volume of revolution, with an arbitrary profile shape:

Photo of a desk lamp

Note the lathe-turned wood stem of the lamp. That’s a surface of revolution, and its interior is a volume of revolution. Any cross section (taken parallel to the table top) is a circle. Thus, if you know a function that gives the radius of the cross section at any height above the table, you can compute the area of the cross section as 𝜋(that)[sup]2[/sup], and with integration, you can find the volume of the entire stem.

Put more formally, if you know a function f(x) that gives the radius of the stem at any height x, then the area of the cross section is A(x) = 𝜋f[sup]2/sup.

Slice the stem into thin slices. The radius will vary from one side of the slice to the other, so use the radius at one side as an approximation. Compute the approx. volume (area times the thickness of the slice). Add them all up, and that’s the approximate volume of the entire stem.

Now, try slicing it into thinner slices instead. Still, the radius will differ from one side of the slice to the other, but not so much. So, using the radius of one side of the slide as an approximation will have less error in it than you had with thicker slices, so you’ll get a closer sum for the total volume. Keep slicing thinner and thinner, and you will zoom in on the true volume. That is the technique of integration.

Next, fill the inside of the stem with your candies. For this step, milk chocolate balls are especially useful. Turn on the lamp. In due time, the chocolate balls will melt, filling the stem with 100% efficiency. As they melt, add chocolate balls in at the top to keep the stem topped up with chocolate. This is an example of the limiting process. In the limiting case, the stem will approach being completely filled with melted chocolate, and the volume of the chocolate in the stem will approach the volume of the stem itself. :smiley:

33

Hey, I’m just the messenger here. Take it up with Him.

34

Here it is:

Note, this is not exactly modern. It’s from 1910. The first link goes to the Wikipedia page, which mentions that there are modern editions available, including a 1998 update by Martin Gardner with some preliminary chapters on functions, limits, and derivatives (what, Thompson didn’t talk about these in his original?), and an added chapter on recreational calculus problems. Allegedly, it doesn’t use the limit concept. In modern Calculus teaching, that would be a major omission, I think.

The modern Calculus textbook that I would recommend is the book by Larson, Hostetler (and maybe Edwards, depending on the edition). I learned from 2nd edition and later 5th edition. I found them very clear and readable. I think I liked the 2nd edition better, maybe just because that’s how I learned it first – they re-organize and re-explain various things with each edition.

They have a web site for their 6th edition. It appears to include the full text, and a interactive lessons, and a message board or chat room. It’s not free. You have to register and I think there’s some price for it. If I had my druthers, I’d just hit up a used book store and buy an earlier edition in the dead tree medium.

35

Hey I’m learning from Khan Academy. I’m just trying to learn some calculus before I can actually do the whole “revolution” thing because I have to actually understand the math behind it before learning something. So right now I’m starting pre-calculus by Khan Academy. You should check it out. It’s an online video series that teaches you math.

36

How is that? Is it free? Is it any good? Does it seem dumbed-down, or not? I’ve heard of Khan Academy but know nothing about it.

I haven’t actually looked at that Calculus Made Easy text that Derleth recommended, so I can’t comment on it myself. But here is a recent post by SDMB member Leo Bloom who is apparently positively impressed.

37

It’s free and yes it seems good. I don’t think it would exactly replace a textbook, but it is teaching me a lot of things like functions and such. I used it a few months ago to review geometry and it was pretty good. It’s a guy named Sal and I think he’s done hundreds of videos so far. They also have practice features. It’s not only for math either. It’s for Science and other related things too, and it even has test preparation. You can check it out for yourself here at http://www.khanacademy.org/ and let me know what you think.

They also actually have specific videos on the whole volume and revolution thing.

38

If that textbook is free, I will try it out definitely. Nothing beats a good text book except for a professor or teacher. I mean Khan Academy is great but not exactly the best for learning. It’s more for review I think and so do others.

The only problem is that would I need to learn pre-calculus first, or can I jump right into this text book and start learning. If I need to know pre-calculus, then I was wondering if the same author has a book called “Precalculus Made Easy”.

39

Wait a minute. Where’s your Trigonometry? There’s gotta be a semester of Trig in there somewhere. You can get a good start in Calculus without trig, but you still need trig sooner or later. (Is it mixed in with the Pre-Calc or something?)

ETA: I don’t specifically remember what we covered in Pre-Calc. IIRC it was included a lot of techniques for proving things, and some advanced Algebra topics like solving higher-order polynomial equations. I think you can get a good start in Calculus without this, just like without Trig. But you’ll still need to learn the Pre-Calc stuff eventually.

BTW: Get a copy of Schaum’s Outline. They have them on many topics, including Analytic Geometry and Calculus. Excellent source of LOTS of extra practice problems, including many fully worked out ones. And good problems, too, not trivial ones.

40

Next good Pre-Calc topic for you to study: (I think I mentioned this already, superficially.)

After you’ve gotten thoroughly proficient with functions, including composition, so you can do it all in your sleep, another seriously useful skill for Calculus is being able to work with INEQUALITIES.

Now, I had some of this in High School Algebra too, but it wasn’t the same. You might have seen problems like: x[sup]2[/sup] + 3x - 15 <= 0
or: 3(x-5)[sup]2[/sup] + 5(y+2)[sup]2[/sup] > 12
and you’re supposed to find the “solution set”. You change it to an equation and graph it. The curve divides the entire plane into two (or more) regions. The “solution set” is one or more of those entire regions, that you’re supposed to draw or shade in. The curve itself may or may not be included in the solution set, depending on how the problem is written. Or, if the equation has only one variable, you graph it and note there it crosses the x-axis. These crossing points divide the x-axis into several intervals. The “solution set” consists of one or more of those entire intervals.

Well, that’s the sort of inequalities we learned in High School Algebra. Those are called “Conditional Inequalities”.

My old Britton & Snively book has a whole chapter on inequalities. It also covers Unconditional Inequalities – statements that are true for ALL values of the variables (or for all values within certain restrictions). The problem is to PROVE it.

( * Pause while I dig out the aulde book from my closet . . . * )

Okay, here’s the first example problem from the book:

(Okay, I stand corrected, the book is from 1947, not 1942 as I mentioned earlier.)

Anyway, there is a whole section on how to prove statements of this sort, and a whole page of practice problems. I studied these on my own before taking Calculus I. This specific topic proved utterly invaluable in understanding the ideas behind the epsilon-delta definition of limits, and in doing epsilon-delta proofs.

ETA: Finally got the spelling of Snively right once and for all.