Tall, Dark, and Mysterious

7/7/2005

Not from the solutions manual of Calculus With Analytic Geometry

File under: This One Time, At Mathcamp, Queen of Sciences. Posted by Moebius Stripper at 1:46 pm.
  1. You’re in the middle of the desert, 2 km north of an east-west river. Your destination is 1 km north and 5 km east of where you stand. You need to stop at the river for a drink at some point in your journey, but it doesn’t matter when. What is the shortest path you can take?

    Solution. Let the river be represented by the x-axis, your starting point be P(0,2), and your destination be at Q(5,3). The shortest path between P and Q that touches the x-axis is equivalent to the shortest path from P to Q’(5,-3). This path is clearly a straight line; properties of similar triangles show that you arrive at the river at (2,0).

  2. A statue is 4 m tall and mounted on a 2 m pedestal. Your eye level is 1.5 m above ground. Where should you stand in order to get the view of the statue deemed “best” by the writers of calculus textbooks - that is, the view such that the angle subtended by the statue at your eye is a maximum?

    Solution. Because I’m lazy and don’t feel like creating pictures, let’s say that the base of the pedestal is at O(0,0), the base of the statue is at B(0,2), and the top of the statue is at T(0,6). Then your eye will be somewhere on the line y=1.5. Construct the circle tangent to y=1.5 that has the segment BT as a chord. Then the point of tangency, P, is such that the angle BPT lies in the circle, while BQT lies outside the circle for all other Q on the line y=1.5. Hence BPT>BQT for all such Q, so you should stand at the P. The Pythagorean Theorem quickly shows that you should stand 1.5 m from the base of the statue.

  3. When widgets are priced at $50 apiece, 100 people buy them. For every $5 increase in price, 2 fewer people buy widgets. How much should a widget cost in order to maximize the widget company’s revenue?

    Solution. Letting x be the number of $5 price increases, the company’s revenue, which we wish to maximize, is R=(50+5x)(100-2x). Equivalently, we may maximize 5/2*R=(50+5x)(250-5x). Applying the AM-GM inequality to this quantity, we have sqrt(5/2*R) < = 1/2*((50+5x)+(250-5x))=150. The revenue is thus maximized when 50+5x=250-5x, ie, x=20. So, twenty $5 price increases, for a price of $150 per widget, gives the maxmimum revenue.

  4. Find the largest possible perimeter of a rectangle whose sides are parallel to the axes and that can be inscribed in an ellipse with equation x^2+2y^2=1.

    Solution. Letting (x,y) by a point on the ellipse such that the rectangle’s vertices are at (x,y), (-x,y), (x,-y), and (-x,-y), we have a perimeter of P=4x+4y to maximize. Apply Cauchy-Schwartz to the vectors (x, sqrt(2)y) and (1,1/sqrt(2)) to obtain P/4=x+y< =sqrt(x^2+2y^2)sqrt(1+1/2)=sqrt(3/2). The perimeter is a maximum when x=2y. Substituting into the equation of the ellipse, x=2sqrt(6), y=sqrt(6), and so the largest perimeter possible is 12sqrt(6).

  5. Two hallways meet one another at right angles. One hallway is 27 inches wide, and the other is 64 inches wide. What is the length of the longest ladder that can be carried horizontally around the corner?

    Solution. The ladder’s length, L, is that of the shortest line segment that passes through the point (64, -27) and is bounded by the axes. The lines passing through (64,27) have equations of the form y=mx-64m-27. The intercepts of such a line are at (0,(-64m-27)) and ((64m+27)/m,0), so we need to minimize L^2=(64m+27)^2+((64m+27)^2)/m^2=(1/m^2+1)(64m+27)^2. Apply Holder’s Inequality with three 3-norms to the vectors (1/m^(2/3),1), (4m^(1/3), 3) and (4m^(1/3), 3). This gives L^(2/3) = ((1/m^2+1)(64m+27)^2)^(1/3) >= (64)^(2/3)+27^(2/3), that is, L>=25^(3/2)=125. So the longest ladder that can be carried horizontally around the corner is 125 inches long.

7/3/2005

Transition

File under: Home And Native Land, What I Did On My Summer Vacation. Posted by Moebius Stripper at 1:23 pm.

Greetings from the happiest place on earth - Mathcamp - where I just realized that I’ll be spending ten consecutive days in the same place, the longest stretch of time I’ll be in one place in two months.

Meanwhile: all of my material possessions (except for the goddamned computer, goddammit) now reside in a storage locker in Vancouver.

Yes, I'm stealing bandwidth. This is what happens when you're at an internet cafe and can't transfer your files. When you live, by yourself, on an island (*) and don’t own a car, such a maneuver requires a lot of careful planning. Rent van; make itinerary of visits to lockers to check them out before committing to any; schedule trips to and from the mainland so as to allow sufficient time to scout out downtown lockers before rush hour and then to transfer materials to chosen locker; return before the last ferry of the day so as to avoid staying overnight in hotel. All of these are subject to last-minute inconveniences: car rental places may be understaffed and deliver your car late, traffic in downtown Vancouver may suck even harder than it did last time, a ferry may run aground and take out twenty-odd boats and result in the cancellation of all service to and from the Island for the rest of the business day. Granted, it’s nice when your ferry schedule makes national news and you don’t have to call 1-800-BC-FERRY to be put on hold for five minutes and be given outdated information anyway (**), but still. No one was hurt, but several hundred people were kept on board for seven (7) hours after the crash, and but for a series of coincidences, I’d have been one of them. Unhurt, but stuck on a ferry with a van full of heavy boxes, kept aboard until after the storage lockers closed for the day, on June 30, a day before Canada Day, when all the storage lockers in Vancouver are closed for the day, and don’t open until after I’ve boarded a train to the US.

Now, to point form, since things are crazy here and I’m being called away every few minutes:

  1. There’s a $400 surcharge for dropping off a van rented on the Island, on the mainland. Since travelling between the Island and the mainland with a car costs $45 each way and takes half a day, and since I can’t imagine employees at a car rental place make much more than $100 for that amount of time it takes to transfer the van back, and since the car rental place has locations in both cities I was travelling between anyway and has a million vans and what’s the difference anyway if one of them stays in Vancouver, this is a very sweet deal indeed for the folks at Budget Rent-a-Car. Consequently, my plan to take the Queen of Oak Bay at 8:30 am on June 30 to Horseshoe Bay had to be changed: to Vancouver on the 29th, back to the Island later that evening, and then back to Vancouver by foot the next day.
  2. Over at the storage locker: a massive truck pulled up along the driver’s side my rented Dodge Caravan, leaving me unable to open the driver’s door. I did what anyone who found herself in such unremarkable circumstances would do: I unlocked the passenger door, with a key, slid over, and stuck the key in the ignition. A siren sounded, and the engine didn’t take. Confused, I called the guy at the rental place. After a false diganosis (leading me to recruit a cabbie from the taxi company next door for a jump, which didn’t work, and being told by said cabbie tht he was being “very generous” by charging me only $10 for the five minutes he was at my service), the rental dude figured out that the car alarm sounds whenever you unlock the passenger-side door first. Because this is how cars are stolen these days: some cunning thief unlocks the passenger-side door, and starts the engine with a key. Except that when the thief’s target is a Dodge Caravan, he’s been thwarted. Thanks, Dodge!

    Related - other, related inane “security” measures I’ve run across lately:

    • I haven’t yet seen a dime of my employment insurance. I made the mistake of using my parents’ permanent address instead of my temporary one, which resulted in my file being flagged. The good folks at EI, while unfamiliar with that downwardly-mobile demographic of which I am a part - twentysomething transients with more education than job prospects, who almost-yearly move to wherever there is work or school - are apparently well-acquainted with that group of unemployed folk who try to defraud the system by filing a single application, under their actual names and social insurance numbers, but listing a different province than the one they’re filing the application from. I may be unemployed, but I’m not stupid, and if I were trying to bleed the system I’d like think I’d be a bit more imaginative, but whatever. So: my file is being transferred from Ontario to BC, which takes four (4) weeks. Feel free to speculate, in the comments, what in the act of transferring a file could possibly take four weeks. Is the EI office hiring? Because I can totally streamline this process for them.

    • If you don’t live in the US, but wish to travel in the US by Greyhound, you’re out of luck. They won’t take your credit card. You can get an American friend of yours to purchase the ticket on their credit card, but then you will have to pay a $15 “gift charge”. Sources of various degrees of reliability tell me that this crackdown on Canadian residents is a consequence of the PATRIOT Act. This seems completely stupid to me, but it’s less stupid than voluntarily turning down business from Canadians, so it’s probably true.
  3. After dropping off my belongings in the storage locker, I made my way to the ferry, forgetting that it’s faster to walk on one’s hands than it is to drive through downtown Vancouver during rush hour. Because of that, combined with the delay caused by the car-alarm fiasco, I missed the second-to-last ferry back to the Island. I didn’t get home until midnight. Exhausted, I slept in. Past seven. Missing the June 30, 8:30 am ferry to Horseshoe Bay.
  4. Which ran aground at 10:10, leaving passengers stranded on board for seven hours.

And that was how a series of little inconveniences allowed me to avoid one big inconvenience, or something. I’m busy and tired and haven’t proofread this, so maybe this post is about something entirely different.

(*) As in, no one shares my apartment, not that I own a private island or anything.

(**) In all fairness, it should be noted that ferry service to and from Vancouver Island is, in general, spectacularly efficient.

6/24/2005

College-level counting

File under: Those Who Can't, Meta-Meta, Queen of Sciences. Posted by Moebius Stripper at 1:22 pm.

[Computer’s still a few time zones away, bike-without-adapter is still primary means of transportation - and I have an apartment to move to the mainland. And I’ll be away, again, this time south of the border, for part of July, so don’t expect anything interesting or substantial in this space for quite some time.]

In the meantime, a discussion topic: how mathematics education went so horribly awry that we now teach this stuff in college. Actually, let me flip that one around: why do we bother teaching this material - integers, fractions, areas of plane figures, basic algebra - in elementary school and junior high in the first place? I know why I think students should learn it, but I wonder if educators and curriculum developers at that level can answer that question, given that basic mathematics is so often so poorly retained. I certainly wonder what insight those educators would offer when confronted with the fact that more and more colleges are offering remedial classes, especially in math, than ever before; that in many cases, it is unwise for us college math instructors to assume that our students have any knowledge of prerequisite material. What sort of subject is grade-school math, that it is absorbed so poorly?

If I were to invent a language with counterintuitive syntax and bizarre vocabulary that bore no relation to that of any Western language, and if I were to teach my invention to a classroom full of schoolchildren, some would excel. Others would do decently. In any event, most, I figure, would pass. But a year later, nearly everyone in my class will have forgotten my crazy language, save a few words that they heard in memorable contexts. Why would they remember it, after all? They have no need for it, and the lessons they took were nowhere reinforced in their day-to-day lives. This is how so many college students see math: as something that they need to learn because their teachers said so, but something that is so poorly connected to the rest of their lives that they have no reason to remember (let alone apply, let alone use) any of it.

At the risk of sounding like an obnoxious student, I’m asking those who teach this material the first time: what’s the point of any of this? Why do we have to learn it? What does it have to do with anything?

…and given whatever those answers are…why is math so seldom worked into any aspect of children’s lives outside the mathematics classroom before they graduate from high school?

6/13/2005

What your physics teacher never told you about electronics

File under: Sound And Fury, Meta-Meta, Talking To Strangers. Posted by Moebius Stripper at 11:31 am.

The Gulf Islands were lovely, just lovely. And I have photos! Postcards! Tales of hitchhiking, prawn delivery, epic seasickness, and more! None of which you’ll get to see or read for a long, long time, because my computer is dead.

“Dead?” I said to the guy at Future Shop, in the way that one asks questions when coming to terms with that chasm between one’s formerly-held beliefs and reality. “I thought it was just the adapter. When I left for my vacation, it was working ok, but the battery was discharging even though it was plugged in. Mind you, when I came back, it sputtered for a bit and then just shut off.”

“Adapter’s fine,” said the technician. “It’s giving me nineteen and a half volts. And your computer won’t even turn on.”

“The hell? It’s only a few months old. And -”

“We’ve had a lot of power surges on the Island in the last few weeks,” the technician recited. “And brownouts. It’s the brownouts that’ll really fry ‘em. Lotta machines in in the last couple weeks, completely dead.”

“Dead,” I repeated.

“Is it still under warranty?” he asked me.

“Hell yeah,” I replied. “And the most important files on it are music, so that’s okay. But - a brownout will just kill a computer? Laptops only, or does this happen to desktops too?”

“We’ve had a lot of desktops in here, too.”

“Is there any way to predict brownouts, or power surges?”

He shook his head. “Nothing you could have done about it. This is not your fault.”

And from there, something about serial numbers and warranties and they’ll send you a box to ship it in and should have it or a reasonable facsimile thereof back to you in a few weeks. A few weeks. I’m going to cart the bloody thing down to the mom and pop computer shop at the north end of town - I still have this rental car for another day - for a second opinion, but barring that, a few weeks.

I’m at the library now, and I’ll be back, because my EI reports need to be filed electronically. Should pick up a few books, while I’m here. I hear people used to read those before there were computers.

6/5/2005

Something for you to do while I vacation yet some more

File under: Queen of Sciences, What I Did On My Summer Vacation. Posted by Moebius Stripper at 9:57 pm.

In a few days I’ll be setting out to visit some more of the Gulf Islands. One of them doesn’t allow cars, and is powered entirely by generators. At least two lack bank machines. One is rumoured to have “honesty stands” where local artists leave their work unattended and trust visitors to pay for what they take. Only one has more than a thousand full-time residents. I may at some point pop into a cafe for a few minutes on a rainy day to avail myself of the dial-up connection on the island’s IBM 486 , but don’t hold your breath. I will be making notes on postcards, so holler if you want one.

In the meantime, some of you may be able to help me with a project I’m thinking about working on one of these days book I have no excuse not to work on now that I don’t have a pesky job to worry about. A bit of background: for the last five years, I have worked at an academic summer camp for mathematically gifted high school students. One of my favourite classes to teach - and one of the most popular among the campers - has been one that I call “Calculus Without Calculus”. Those of my readers who know me in real life are well aware that I…well, I don’t hate calculus, so much as I think that calculus doesn’t need me to love it. Calculus gets more than enough attention in the thousands of high schools and universities that inflict it upon every other student that passes through their doors, the overwhelming majority of whom don’t learn it properly and wouldn’t ever use it again even if they did. Nevertheless, calculus is a natural choice for students who have not learned to think mathematically: for all the terror it strikes in students’ hearts, it’s one of the easier branches of mathematics to reduce to mindless algorithms in a low-level course. Need to maximize some quantity? Set a derivative to zero, and solve. What were we trying to do again?

Calculus Without Calculus is a collection of methods of solving typical calculus problems without taking a single derivative. The two main methods involve inequalities, or exploiting geometric properties of figures, particularly symmetry. The old “maximize area with given cost of rectangular fence” problem? You can complete the square, take a derivative - or you can apply the AM-GM inequality. The question about getting the best view of the statue on a pedestal, that appears in the chapter on inverse trig functions in every single calculus book? Solvable using elementary circle geometry that could be found in every grade ten math text before it was decreed that geometry should no longer be taught. There are tons of these. I know of three calculus-free methods of finding tangents to ellipses: one using transformations of circles, another using the Cauchy-Schwartz inequality, and one using projective geometry. A few months ago, I was reading some fiction on the Vietnam War - One To Count Cadence, by James Crumley - and one of the characters mentions in passing that he was able to solve the ladder-around-a-corner problem (you know the one) without calculus. I struggled with that for a long time before a camper provided the key insight. A quick application of Holder’s Inequality, and the result falls right out. (Except that now I’m trying to recreate it. Damn; this is going to keep me up again.)

I’m sure there’s a lot more to say on the topic, and I’d like to write a (short) book on the topic. What I’m looking for: book recs. In particular, recommendations for good books on problem-solving, which tend to spend a lot of time on funky inequalities. I’m especially interested in geometric and otherwise intuitive proofs for the old standbys (AM-GM, Holder’s, C-S…) , and lots of examples - both of single variable problems that one would see in calculus texts, as well as multivariable ones that are a lot easier to solve without calculus. One of my favourite books of this sort is Problem Solving Through Problems, by Loren Larson, the talented mathematician and educator who first introduced me to this stuff. If you’ve got any recs, or anything even slightly pertinent, I’d love to hear them - post them below so I’ll have something to check out when I return to this big island.

My other reason for posting this, of course, is that now I’ll feel horribly guilty if I don’t actually have anything to show for this in a few weeks.

And, if [above] isn’t your thing, perhaps the Phallic Logo Awards can keep you busy for the next week. (What’s this you want, a better segue? Here: Galiano Island.)

6/3/2005

‘You have the right to remain silent. So SHUT THE FUCK UP.’

File under: Miscellany, Meta-Meta. Posted by Moebius Stripper at 10:43 am.

There’s no shortage of blogs by cranky educators out there, but to the best of my knowledge, there aren’t many by public defenders. Which bodes well for my traffic, because the occasional “If you’re going to try to convince the department head that you know the material, perhaps you shouldn’t tell your instructor that you haven’t done any work in three weeks” doesn’t hold a candle “If you have some miscellaneous drug charge, think twice about clothing with a marijuana leaf on it or a t-shirt with the ‘UniBonger’ on it.”

[via mythago]

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