Dirtyknees...

"Why is this dripping?" is never a good question.

Mathletics

A cranberry bog fills with water in nine hours. After the bog is full, a dike is opened and it drains in 11 hours.

How long will it take for the cranberry bog to fill if the dike is open from the start of the filling process?

(Hint: See comment in Briefblog.)

Thanks Shelley. Thanks Franz.

10 Comments:

At 5:09 PM, Anonymous said…

dude, considering that you know a large chunk of your readership is a bunch of geeks, why, why do you do this?

would you like an algebraic solution? a solution using nothing more than aritmetic? a solution that appeals to the hyper-real numbers?

c'mon, hook me up
At 5:16 PM, c. said…

I like hyper real numbers. Give me those. They remind me of the "hyper color" shirt I owned in 1989.
At 2:14 AM, Anthi said…

No solution yet? I'll solve it in the morning if no one else has by then.
At 6:10 AM, c. said…

The dragon has yet to be slayed.
At 9:06 AM, Anonymous said…

Let's begin with a brief discussion of the hyper-real numbers.

That is, WFT are they?

Here's where things get fun. See, the real numbers are can be thought of as an unbroken-line. They strecth off to infinity in both the positive and negative direction. How close together are the real numbers you ask?

Well I say, if you imagine zooming in on a small portion of the real line, no matter how powerful your 'zoom-feature' you will never be able to see a break.

Some comparisons: if you take an atom of, well, whatever, Hydrogen and zoom in on it. Eventually you'll realize that the space between the electron and the proton would take about 1000000 electrons lined up to cross. There is even space between the sub-atomic particles that make up protons.

So, the real line--so tightly packed that, in effect, you can't tell the difference between 1 and .99999... or between 1.000...001 (if we could actually write that number down).

Now, that I've just spent time convincing you that there are no spaces in the real line, let's talk about the hyper-reals.

Imagine dividing 1 by infinity and you get a really small number, maybe even a hyper-small number (infinitesimaly small some might say). If you take 1 and subtract that really small number you will land squarely between 1 and .999999... and that number, the one between 1 and .9999... sir, is a hyper-real number. There's actually a whole lot of them in between 1 and .9999...

This is what the hyper-real numbers are.

You can completely re-formulate calculus using the hyper-real numbers and thereby eliminate the use of limits in the whole deal. It's kind of fun. There are a couple of 60's texts that make use of them, one is actually a calc-text.

We'll continue this after I've finished writing chapter 3 of my dissertation. Anyone on here know anything about Kalamazoo?
At 11:36 AM, Anonymous said…

so, math guy,
i read your post about hyper-real numbers, and i still can't conceptualize the idea. your analogy to atoms doesnt make sense to me because atoms are part of actual reality - sorry - matter, that is. while math for the most part are representations of the matter, and its interactions. so in the case of the infinetesmally small gap between 1 and .99999etc. that you described why couldnt there be a real number vs. a hyperreal number, since math is a logical concept.
so i guess my question is also if this number is hypereal, then infinity must be also, right?

-oso loco
At 1:55 PM, c. said…

We have a winner, er...

the dragon hath been slayed.

THREE CHEERS FOR SIR ANTIPLATYPUS!!!

HOORAY!

YEAH!

YIPEE!

49.5 hours is the correct answer to the problem.

This problem even stumped a good friend who aced the math portion of his GRE. It must be mentioned however that said friend was working on three hours of sleep and a greasy hospital breakfast.

Great job my Wilsonville friend. Thanks for the in depth analysis as well. Evaporation coud play a significant role in some environments, but I forgot to mention this bog is found in a temperature controlled vacuum.

You are a knight and a honored mathlete.
At 3:24 PM, Anthi said…

I feel honored.

I spent weeks doing grueling mathematical calculation. I checked out hundreds of books from the library. I learned greek so that I could read the original treatises upon which modern mathematics is based. I consulted with various professors of mathematics at some of the top universities in the world. Finally, I meditated for days upon the very nature of mathematics itself, upon the very nature of symbols, and in a bright flash of pure enlightenment, I felt as though I understood the universe in its entirety. It was then that I discovered the answer.

Sadly, my enlightenment did not last very long, and I have lost much of the wisdom that came with enlightenment. I plan to retire to my cave of solitude in order to begin the process again. Perhaps this time I can hold onto enlightenment for just a little longer.
At 10:24 AM, Anonymous said…

to follow-up to my digression on the hyper-reals:

yes, in this made-up-world, infinity is a number and you can calculate with it.

why would we ever use the hyper-reals given that they seem to be so completely artificial (in fact, many leading mathematicians refer to them as "bunk")?

Well, the answer my friends is because they make certain calc-level and above calculations rather nice. They don't do crap for algebra.

Regarding other constructions that seem(ed) similarly ridiculus, I would now like to draw your attention to the concept of the square root of negative 1. Specifically, we call this the imaginary number and denote it as i.

For the first 200 years after someone said, "hey, we should have a number that does that" most mathematicians in the world said something worse than 'bunk' but since they said it in Italian, French and Chinese we pretty much ignored them, oh, and they're all long dead. In fact, i was first proposed as a place-holder for an intermediate calculation for solving 3rd degree polynomials by Cardano in the 16th century. You will most certainly never, in the real world, find anything that represents the square root of negative one. (Really, you'd be hard-pressed to come up with 'normal' stuff that represents negative times negative equals positive)

So, why has it survived? Well, because upon the creation of electrical engineers someone decided that i makes all of the important calculations that they do so much better, and, in fact, lead to the creation of the integrated circuit and the very screen upon which you are reading this incredibly geeky text. The moral: just because you can't see it doesn't mean that it can't do some useful stuff.

Actually, as far as anyone can tell, they hyper-reals do nothing useful other than make a few calculations easier, or, and make for nice party-amusement when you're at a party filled with science and math nerds.
At 11:22 AM, c. said…

I like oatmeal.