So I encountered what I think is a horribly wonderful (and did I mention horrible?) problem. It took me forever to solve, because I kept getting different solutions and I didn’t know which was right and which was wrong and where my errors might have come from. But in the end, I think I finally got it right. The error was a silly sign error which made everything nice.

Here’s the problem [1]:

A length of sheet metal 27 inches wide is to be made into a water trough by bending up two sides as shown in the accompanying figure. Find and so that the trapezoid shape cross section has a maximum area.

I don’t want to tell you what I got for the answer, in case you want to try it out for yourself, but I will tell you this: I was able to get a maximum area of  square inches.

(And if you are banging your head against a wall, and need a solution for peace of mind, shout out in the comments. I’d be happy to type out my solution. I just don’t want to spend the time if no one cares.)

Update: I had  posted as the answer, but again, I noticed a second silly sign error. I have never made so many algebraic errors on an problem as I did with this one.

[1] From Anton’s Calculus, 8th edition, Chapter 14, Section 8, Number 44

The New York Times has an article about mirrors, which had a few passages which blew me away. I think of mirrors as relatively simple entities, until they get curved and start flipping images upsidedown… depending on where you are looking at the mirror from. But no, mirrors are interesting as simple, flat creatures too:

In a series of studies, Dr. Bertamini and his colleagues have interviewed scores of people about what they think the mirror shows them. They have asked questions like, Imagine you are standing in front of a bathroom mirror; how big do you think the image of your face is on the surface? And what would happen to the size of that image if you were to step steadily backward, away from the glass?

People overwhelmingly give the same answers. To the first question they say, well, the outline of my face on the mirror would be pretty much the size of my face. As for the second question, that’s obvious: if I move away from the mirror, the size of my image will shrink with each step.

Both answers, it turns out, are wrong. Outline your face on a mirror, and you will find it to be exactly half the size of your real face. Step back as much as you please, and the size of that outlined oval will not change: it will remain half the size of your face (or half the size of whatever part of your body you are looking at), even as the background scene reflected in the mirror steadily changes. Importantly, this half-size rule does not apply to the image of someone else moving about the room. If you sit still by the mirror, and a friend approaches or moves away, the size of the person’s image in the mirror will grow or shrink as our innate sense says it should.

What is it about our reflected self that it plays by such counterintuitive rules? The important point is that no matter how close or far we are from the looking glass, the mirror is always halfway between our physical selves and our projected selves in the virtual world inside the mirror, and so the captured image in the mirror is half our true size.

Rebecca Lawson, who collaborates with Dr. Bertamini at the University of Liverpool, suggests imagining that you had an identical twin, that you were both six feet tall and that you were standing in a room with a movable partition between you. How tall would a window in the partition have to be to allow you to see all six feet of your twin?

The window needs to allow light from the top of your twin’s head and from the bottom of your twin’s feet to reach you, Dr. Lawson said. These two light sources start six feet apart and converge at your eye. If the partition is close to your twin, the upper and lower light points have just begun to converge, so the opening has to be nearly six feet tall to allow you a full-body view. If the partition is close to you, the light has nearly finished converging, so the window can be quite small. If the partition were halfway between you and your twin, the aperture would have to be — three feet tall. Optically, a mirror is similar, Dr. Lawson said, “except that instead of lighting coming from your twin directly through a window, you see yourself in the mirror with light from your head and your feet being reflected off the mirror into your eye.”

This is one partition whose position we cannot change. When we gaze into a mirror, we are all of us Narcissus, tethered eternally to our doppelgänger on the other side.

What a great thing to know… NYT graphic below:

Today there was an Earthquake in Southern California. A NYT article said:

The quake, estimated at 5.4 magnitude (reduced from an initial estimate of 5.8), was centered 35 east of downtown Los Angeles in Chino Hills, just south of Pomona in San Bernardino county. It was felt as far east as Las Vegas and as far south as San Diego.

My first reaction, a question: how much of a difference was there in terms of the seismic energy released at the epicenter of the estimated earthquake versus the actual earthquake? How off was the esimate? I know that the Richter Scale is logarithmic, so the answer would be:

The estimate was over 2.5 times off.

But I realized: I know very little about the Richter Scale and how earthquakes are actually measured. How could an initial estimate be so wrong? I’m going to use this post to explain what little I’ve pieced together from the internet.

Jump on below!

(more…)

My sister told me this physics problem, and I have no idea how to solve it. It goes like this. There are two identical balls, one lying a distance x from the edge of a frictionless table, and the other being held up in the air, a distance x from the edge of the table. (See diagram above.) They are connected by a massless rope.

At time t=0, the ball hanging in the air is dropped.

Do you see what’s going to happen? That ball will start to fall down and approach the table, while the ball sitting on the frictionless table will be pulled toward the edge.

Two questions, both of which I can’t answer:

(1) Which will reach the vertical plane of the edge of the table first? (In other words, will the ball on the frictionless table fall off the table before the falling ball hits the side of the table? Or vice versa? See my diagram below to see what the estimated path of the falling ball will look like. It curves towards the table, but also the distance from the corner of the table to the falling ball will increase as more time passes.)

(2) What is the equation for the path of the falling ball? (Again, see diagram below.)

It seems I was scooped.

In my ongoing obsession with the law and math (see post I and post II here), I learned that for many years, all the supreme court justices shake hands before each meeting.

The “Conference handshake” has been a tradition since the days of Chief Justice Melville W. Fuller in the late 19th century. When the Justices assemble to go on the Bench each day and at the beginning of the private conferences at which they discuss decisions, each Justice shakes hands with each of the other eight. Chief Justice Fuller instituted the practice as a reminder that difference of opinion on the Court did not preclude overall harmony of purpose. (here)

It still goes on today. There are 9 justices. A natural question: how many handshakes?

The “handshake problem” is a common problem in math classes. It’s equivalent to the problem of calculating the number of diagonals of a polygon. (Each vertex is a person; each line — including the edges of the polygon itself — represents a handshake.) It’s equivalent to adding the numbers .

I was thinking it would be a good hook for a class where this problem was presented. Better than:

Assume we have n people at a dinner party who all don’t know each other. They all introduce themselves to (yawn) each other… (yawn)… and shake hands. How many (yawn) handshakes are there?

In the same sense that that’s not a true-to-life problem, neither is the supreme court handshake problem. But it’s a hook.

Of course, I looked it up to see if anyone else had thought of it (likely) and guess what? As I said: I was scooped.

I was so enthralled with Judge Sand’s city of Yonkers case that I couldn’t help but do a little more research on him. And lo and behold, nary a page in from a google search, I found another case that deals with math!

The setup

Henry Barschall, working for the American Physical Society and the American Institute of Physics, investigated the cost of science journal subscriptions for libraries. Even then in 1986 (as we hear is still the case now), science journals were increasing their prices — for a variety of reasons. And science journals were on average more expensive than journals in other disciplines (e.g. in 1985, the average yearly subscription for a science journal was $328, while for philosophy, it was only $47 — here).

The survey was done and ready to be published in Physics Today. Barschall outlined his methodology here. The intial results which were published in 1986 are below:

Note that for the physics journals, the American Physical Society and the American Institute of Physics were the most cost effective (0.7 cents/1000 characters), while Gordon and Breach were the least cost effective (31 cents/1000 characters). For this case, it’s important to note who sponsored the study! (The same organizations who came out the “winners.”)

An expanded survey, published in 1988, took into account not only the price, but the “impact” (or importance) of the journal… The metric for the impact was the average number of papers which cited an article in the journal two years after its publication. The more “impact” the journal had, according to the study, the more it mitigated a higher cost (hence: a ratio was created).

Again, APS and AIP were near the top of the list, while Gordon and Breach were way down at the bottom, with a cost/impact ratio more than twice the next publisher.

Gordon and Breach sued the APS and AIP. Judge Leonard Sand heard the case, and summarized it as follows:

This action, a dispute between publishers of scientific journals, raises an issue of first impression: whether a non-profit publisher may be sued for false advertising under the Lanham Act for publishing comparative surveys of scientific journals which, through the employment of a misleading rating system, rate its own publications as superior.

[…]

These articles, G&B alleges, constituted the start of a “continuous promotional campaign” waged by AIP and APS against them with the aid of Barschall’s survey results. First, G&B alleges, AIP and APS distributed “preprints” of the 1988 survey results to librarians (the primary purchasers of scientific journals) at a conference in June 1988. Since that time, G&B alleges, defendants have continued to disseminate the results of Barschall’s surveys to prospective purchasers through press releases, letters to the editor, “electronic mailings,” and meetings with librarians (“secondary uses” of the articles). G&B states that its attempts to reach an accommodation with defendants have been fruitless, and that it has been forced to resort to the courts. After filing a series of legal actions in Europe claiming unfair competition, and failing to receive satisfaction in Swiss, German, and French courts, G&B brought this lawsuit in September 1993.

G&B contends that the articles are promotional materials cloaked in the deceptive guise of “neutral” academic inquiry, and thus constitute misleading advertising under the Lanham Act, 15 U.S.C. § 1125(a), and comparable provisions of New York General Business Law. Barschall’s studies, G&B contends, far from being neutral, in fact constitute a “cynical promotional campaign” by AIP and APS to deceive librarians and other consumers of scientific journals into thinking that their journals have been “scientifically” proven to be of superior value.

Judge Sand’s judicial opinions, the trial transcript (really interesting stuff! I swear!), Barchall’s papers, and other related material have been put on this site at Stanford for you to read.

Eventually in 1997 (yes, justice takes a long time to render), Judge Sand made his final ruling:

Barschall’s methodology has been demonstrated to establish reliably precisely the proposition for which defendants cited it — that defendants’ physics journals, as measured by cost per character and by cost per character divided by impact factor, are substantially more cost-effective than those published by plaintiffs. Plaintiffs have proved only the unremarkable proposition that a librarian would be ill-advised to rely on Barschall’s study to the exclusion of all other considerations in making purchasing decisions. This consideration in no way makes Barschall’s study or defendants’ descriptions thereof false, and accordingly judgment is granted to defendants.

And the case is over!

Could this be useful in a math class?

I don’t know. I honestly can’t see myself using it next year, except maybe for a quick classroom discussion — a few “do nows” where each day I introduce more and more of the case and the evidence, and we talk about the math issues it raises…

1. Before presenting the data, it could lead to a discussion of how one might measure the cost-effectiveness of journals, and different ways students could imagine calculating them.
2. Does Barschall’s method of measuring impact make sense? Does taking a ratio of cost per thousand characters to impact work? Why divide? (That will be hard for my students to get.)
3. A baby project: students could go to the school library and get data about various online databases that the library subscribes to, and try to determine whether subscribing is cost-effective for the school. Or at least, the class could try to brainstorm out what sorts of data we would want (e.g. the cost of the database annually, whether — and if so, how quickly — the cost of the database has been increasing, the number of times the databases are accessed each day, the amount of information the databases contain, etc.)But honestly, this seems like it isn’t really worth it. It would make for an interesting independent study, though.
4. A discussion about whether the survey was ethical or not, merely by virtue of the fact that those producing it were players in the game (non-profit players, but players none-the-less). [A similar discussion would be whether drug companies should produce their own studies of drug effectiveness.]
5. Ideas for how one might present the data visually?
6. Should a court (even with scientific experts being called in) be the final arbiter in whether a mathematical study/survey was sound or not? If not, who should be?

THE PROBLEM IS HERE!

In the Quach family, there is an age-old tradition that has been intact since the 1500s. In this tradition, every member of the Quach family receives one lottery ticket for his/her first 25 birthdays.

Based on empirical data collected over the years, it was determined by the family Mathematician that the probability of a Quach family member winning the lottery at least once during that 25 year stretch is 211/243.

Assuming the above statement is true, what is the probability of a Quach family member winning the lottery at least once during any 5 year interval? (Assume that lottery wins are completely independent of each other).

So the probability that a member of the Quach family does not win the lottery in 25 years is 32/243. Since winning each year is independent of any other year, we know:

P(not winning any year)=P(not winning 1st year)*P(not winning 2nd year)*…*P(not winning 25th year)

And since the probability of not winning each year is the same, we get:

P(not winning any year)=P(not winning in nth year)^25
32/243=P(not winning in nth year)^25
P(not winning in nth year)=(32/243)^(1/25)

Now we need to find the probability of winning at least once in a five year interval, which is actually 1-P(not winning in 5 year interval).

P(not winning in 5 year interval)=P(not winning in 1st year)*P(not winning in 2nd year)*…*P(not winning in 5th year)

P(not winning in 5 year interval)=P(not winning in nth year)^5

P(not winning in 5 year interval)=(32/243)^(1/5)=2/3

Since we want to find P(winning at least once in 5 year interval), we know that that will have to be 1/3.

I want me some of them odds.