Probability in the Pool: August 19, 2013

When my mom had a catastrophic stroke, I returned to my home town and found a pool in which to swim.  I swam mostly alone in a pool that was 50 meters by 25 meters.  Sometimes the lane lines were set to allow swimmers to swim "short course" (25 meters) or "long course" (50 meters).

On one of the short course days, I was alone in the pool.  There were 16 empty lanes.  I chose lane 8, one of the middle lanes.  I swam down-and-back (50 meters) in about 45 seconds.  This became a deliciously long swim as I moved back and forth alone in this pool with my own personal lifeguard.  Then, suddenly, I was jerked out of my fraction-calculating delerium (refer to previous post about Fractions in the Fast Lane) when waves overtook me.  A man dropped himself and his large belly covered in fur into the lane next to mine at precisely the moment I was at his end of the pool.

Ah, I thought.  A probability problem.   What is the probability that he would (a) choose the lane next to mine and at the same time (b) choose to enter the water during the roughly 7 seconds that I am vulnerable to the tsunami he created at the near end of the pool.  And is this probability small enough that I should think this individual inconsiderate?

On one of the long course days, I was again alone in the pool, now well spoiled and feeling like the Queen of Sheba in her own blue-glass lake (with lane-lines and a black line at the bottom.)  There were 9 lanes. I chose the middle lane.  I would swim 50 meters to the far end and 50 meters back, giving me new distances and fractions to consider.  I swam the same rate as in the short course setting in the pool.  Again, as I swam a deliciously long swim with new numbers flowing along side of me, another man, a more narrow man, entered the pool.  Now then, many of us in this world like to be individuals; we value our ability to make choices and to be unique.  This man decided to swim a uniquely different way in this pool.  He wanted to swim 25 meters, not 50 meters as the pool was set.  So he did, weaving his way across the pool, below the lane lines and intersecting my path perpendicularly.  My interest became piqued.  As I swam, I watched the red clocks surrounding the pool and observed that he swam breast-stroke quite regularly, finishing 50 meters (down and back, under the lane-lines) in about 100 seconds.

Ah, I thought. Another math problem. If we both start at the same place (let's say a corner of the pool, just to make this conceptually more straight-forward) and swim in paths perpendicular to each other, when (if at all) will we collide?  If we both leave one end of the pool at the same time and swim at our own constant paces, when (if at all) will we collide?  If a moment during our infinite-length swim is chosen randomly, what is the probability that we will be in a collision at that moment?  Consider that it would be about 7 seconds for me to be in his way; he'd be in my way for 1/9th of the way across the 25 meter pool.

TPC: Waves in the Pool

I swim at a constant rate.  Well, close enough; assume I swim at a constant rate, then we'll adjust that later in this blog.  That rate is roughly two meters for every second.  Well, ok, that's a little quick for this ol' lady, but those numbers work for us. So pretend.

Let's start by thinking of what a graph would look like as I cruise at my theoretical constant pace of two meters per second back and forth in a pool that is 25 meters long.  Let the x axis (independent variable) be time, measured in seconds, and let the y axis represent meters away from the wall, or side, of the pool.  Let's start our stopwatch at the exact time when I leave the wall and head to the other side of the pool.

How does the graph appear if I adjust my pace to 1 meter per second?  How about to 3 meters per second?  1/2 meter per second?

These kind of "waves" are not quite a sine wave. Wikipedia categorizes the waves as triangular:

http://en.wikipedia.org/wiki/Sawtooth_wave.

But there's more. Lots more.

How would the graph be different if I swam my 2 meters per second pace in a pool that was 50 meters long?

How would I need to swim differently to make the graph look like the other waves that the wiki pictures on the bottom right?  Which ones are not possible?  Spend some time with, of course, the sine wave and describe how I'd need to swim so that my distance from the wall would be a lovely, smooth sine wave.

So you've already seen that I do some interesting math while I swim. I can solve all the world's problems when I swim, you know. Now what happens if I'm swimming in the pool that's 50 meters by 25 meters.  I swim the long way at a constant 2 meters per second and another swimmer decides to swim the short way, perpendicular to my path and swimming a constant 1/2 meter per second. (No kidding, someone actually decided to do this last summer.)  I swim in lane one as does he (well, we don't but pretend we do) so we start at the same time and in the same corner. Because we are all thinking perfectly and precisely mathematically, we imagine triangle waves, but let's say we are so far advanced that both the other swimmer and I slow down and speed up appropriately so that the graphs of our distances from the wall with respect to time form perfect sine curves.  You've already worked out  how we adjust our speeds to make perfect sine curves, of course, because the previous paragraph asked you to do this.

So the bigger question is, assuming we can occupy the same spot simultaneously or that I can duck directly below the surface while maintaining my lovely sine curve pacing, how often do our paths intersect?

For your blog, either show graphs like I've described or take my idea a little further, maybe by answering the questions I pose, OR find your own application of a sine curve.  Where do YOU see it in the real world?

9%.

Fun to bike down but a workout to bike up, a 9% grade earns a failing grade in my gradebook. 

Outside of Otis, Massachusetts is a road with a very steep hill.  Put in neutral, our standard transmission car just cruised down the hill. On the way up, first gear was the way to go, so to speak. The rhomboid sign reported a 9% grade (“Test your brakes,” it warned). A biker was huffing her way up the slope while a second simply sailed down the hill.  What’s the 9% mean?  If 60% is passing and 90% is an “A,” what’s 9%? Doesn't seem like much; why the big deal on that hill outside Otis?  Folks seem to always aim for 100%, but that would be suicidal in an automobile or bicycle and certainly not preferable.  We, as humans, do our best to categorize (think: Kingdom, Phylum, Class, Order, Genus, Species or better yet square, rhombus, rectangle, parallelogram, trapezoid, quadrilateral); it seems we have categorized slopes (or grades) of hills as well. 

Wales has a road with a 25% slope; I-70 into Denver from the west has a cool 6% grade.  A handicap ramp has to be an inch vertically for every foot horizontally.  Are these ideas related? 

Your mission is to understand what these numbers mean and how engineers have come to categorize the grade of a road, ramp, or slope.  Nice word there, by the way, “Slope.” 

http://en.wikipedia.org/wiki/Grade_(slope). 

Yep, good ol’ Wikipedia actually has a description that works for us.  It may seem a little dense and might take some slower reading than, say, Ted Geisel’s stuff,  but it’s got all the ideas you need.  In the wiki, there are triangles, a protractor shape, a trigonometric function, and some other very familiar words.  Put the pieces together in your blog and you’re set for the week’s blog assignment. (Be sure you take out the irrelevant ideas for "grade" in my post -- this is meant to have nothing to do with the grade you get in class. That's a joke.)

More specifically, the assignment for both TPC and GA2:  the grade of the road has everything in the world to do with a trig function.  Which one? Why? Explain.  Use roads that you've seen or know about or find on line.  There's a couple different standards for handicap ramps (businesses vs private homes); find those if you'd like.  Go bananas on this one -- where else do you hear about grades?  What about the "angle of repose"?  What's that?  What about "railroad grades"?  Choose something that interests you; don't feel as though you need to cover absolutely everything, but DO cover the idea of what a "grade" is.   If you are one of those folks in GA2 fascinated by the number theory topic we touched on (Pythagorean Generators), you can choose to write on that instead of this whole idea.