Schlong Division

10 06 2011

WARNING: THIS POST IS ABOUT A RECENT EPISODE OF SOUTH PARK. AS SUCH, IT IS MORE OR LESS A DISCOURSE ON THE MATHEMATICS OF THE MALE GENITALIA. IF YOU DON’T WANT TO READ ABOUT THAT, LEAVE NOW.

I love South Park and I love math. So I was pretty stoked when I saw the new episode entitled “T.M.I.” last month. Quick synopsis: Those goofballs in Colorado declare that the size of a man’s–ahem–member can not be adequately described using just one number (length), and so Randy Marsh introduces an equation (video here) for adjusted length, or T.M.I:

Now, this equation should set off some alarms for any math dork because when you look at the units, it makes no sense.In the numerator, we have (Lxd), which gets you units of length squared. We also have the term W/G, weight over girth, which gets you units of weight per unit length. And then you add length squared to weight per unit length? That makes no sense! And don’t get me started on the fact that they were using kilograms as their unit of “weight!” (Kilograms are mass, not weight) AND they mixed imperial units (inches) with metric units (kilograms)!

Ignoring the unit fiasco, let’s look at what this equation actually says. I think for the sake of the discussion, it would be easier to rewrite it like this:

The first thing I notice here is that the T.M.I. increases with the product of length and diameter. This makes sense- a girthy wang can make up for shortcomings in the length department, so it’s important to consider both. In the second term, we have weight divided by girth- a sort of quasi-density measurement. This term would tell us how much the section under an arc of a given length would weigh. So we see that a higher density leads to an increased T.M.I. So far the equation kind of makes sense if you ignore the units. But then we come to the troubling part: dividing by the angle. The cartoon fails to describe just what it means by “angle of the tip.”

But all this is minor compared to the epic math fail that comes later in the episode (video here). The doctor says that when somebody is “consistently angry, or always finding new reasons to be angry, it means they have a very very very very small dick.” This statement implies that there is a negative correlation between penis size and anger. And he writes the following equation on the board:

where L=length, W=width, and M=mass. The equation shows a positive correlation between penis size and anger,directly contradicting what he says while referring to the equation. Like the Chewbacca Defense, it makes no sense.

Overall I was disappointed by the nonsensical pseudo-math in the episode. The equations they use are clearly just slapped together, designed only to sound complicated and get a cheap laugh. I get pretty stoked when cartoons put thought in to their math jokes- like the Simpsons’ gag about Fermat’s Last Theorem or the Futurama episode that introduced an original theorem.

It’s not all doom and gloom, though. South Park did seem to accurately use the term “yaw.” (In the interest of not having to draw diagrams, I’ll just leave it at that.) And in the end, it was a pretty good episode, with the South Park guys doing what they do best. They used the town of South Park as an allegorical microcosm of an issue that’s going on in the world, and took the ideology behind the issue to its logical extreme–with hilarious results. So I can forgive them for their sloppy math.





Ferment’s Last Beer-em

5 06 2011

I enjoy a good pint of beer (or root beer, for my younger readers) as much as anyone. Which is why I was intrigued when I saw this article: http://www.avclub.com/denver/articles/of-legal-gauge,27021/ It’s about a guy who, tired of getting short poured at his local drinking establishment, created a gauge to measure the volume of beer in his pint glass based on the height of the beer. Reading the article made me curious about how one would determine the volume of beer in one’s pint glass.

As I write this, my pint glass is this full:

Enjoy responsibly.

The casual observer might look at the glass and say, “Three quarters full! Oh yeah!” But are you ready to have your mind blown? Here it is: That glass is about half full. That’s right. Half. The reason has to do with the shape of the glass: it’s conic. Since the top of the glass is wider, it holds more beer than the bottom of the glass.

I guess the logical place to start here is to calculate the volume of beer in the pint glass. There are a couple ways to do it.* In the interest of being a responsible citizen, I’m going to use calculus because if you know calculus you’re probably at least close to legal drinking age. If you don’t know calculus, you should skip down to the paragraph that starts with “If you don’t know calculus, rejoin me here.”**

The shape of the pint glass lends itself to cylindrical coordinates, so let’s do a volume integral in cylindrical coordinates. We’ll assume the edges of the glass are straight, i.e. the glass is the shape of a cone with the point chopped off. I’ll draw half the glass on the cylindrical coordinate axes. I’m only drawing half because in the integral I’m going to integrate it from 0 to 2π (alternatively, I could draw the whole thing and only integrate from 0 to π):

In case it’s not clear from the drawing, rtop is the inner radius of the glass at the top, and rbot is the inner radius of the glass at the bottom. L is the length of the part of the glass that can be filled.

Recall the general formula for a volume integral in cylindrical coordinates:

We need a function for r, which is just a straight line:

Where z is the height of beer in the glass. I’ll skip a few steps here and just give you the solution to the integral.

If you don’t know calculus, rejoin me here. I’m giving you a handy formula that can be used at any watering hole to see how much beer you’re actually getting, assuming a conic pint glass: (Note: this formula is unitless, so whatever units you plug in are the units you’ll get out. You may want to convert to ounces or milliliters to get a meaningful number.)

(Fun fact: If you set z=L and rbot=0, you get the formula for volume of a cone of height L. Neat-o!)

So now that we have the equation, let’s plug and chug (pun intended). Let’s calculate the volume of my pint glass, which as far as I know could be identical to any other conic pint glass. I measured the following dimensions, which may not be exact but they’re pretty close:

Using the above formula, it looks like my glass has a volume of 29.57 cubic inches, or 16.4 fluid ounces. A pint is 16 oz, so this is close enough given the rough measurements that I took. If I solve the equation for several values of z and graph it, it looks like this:

See how the slope increases as the glass gets fuller? This tells me that the volume increases at a higher rate as the glass gets closer to full. In other words, the math tells us what we already know intuitively: most of the volume is at the top of the glass! That’s because cross-sectional area increases as a function of the radius squared (Area = pi r^2) Therefore, since radius increases linearly with height, cross-sectional area increases as a function of the height of beer squared! Therefore, missing an inch of beer at the top of your glass is a way bigger deal than if you were missing an inch of beer at the bottom of your glass. In fact, you can see from the chart that an inch of beer missing from the top means you’re missing a full 25% of your pint! So next time the bartender pours you a beer, make sure he pours it to the top. And that’s how math can make you a better beer drinker.

An inch from the top – almost full? No, only ¾ of a pint!

*You should also be able to derive a volume formula not using calculus but instead based on simple high school geometry. (V=1/3bh, and some sines and cosines) I haven’t done it, but I’d love to see the derivation if anybody does!

**If you want to learn calculus, check out http://www.khanacademy.org/#precalculus as a place to start. The site has excellent, completely free, pre-recorded lectures on pretty much the entire standard math curriculum from arithmetic all the way up to differential equations. Calculus was a revolutionary way of looking at the physical world when it was discovered. It is still amazing and extremely useful.





I, for one, welcome our new leviathan overlords.

3 06 2011

I was watching an old episode of The Simpsons the other day when I heard the following quote from cartoon anchorman Kent Brockman:

Kent Brockman :  Did you know that thirty-four million Americans are obese?  Taken together, that excess blubber could fill the Grand Canyon two-fifths of the way up.  That may not sound impressive, but keep in mind it is a very big canyon.

That statistic was made up, but it made me wonder how much excess blubber there really is in the US. In other words, if everybody who is overweight got liposuctioned down to a healthy weight right now, how much fat would we have to dispose of? Every day there’s a new report telling us that Americans can’t stop cramming fast food and donuts down their gaping pie holes. Just how much of those maple glazed treats and KFC Double Downs are stuck to our thighs and bellies right now? I looked up a few statistics.

From a 2008 CDC report:

The average adult male is  69.4 in tall.

The average adult male weighs 194.7 lb.

The average adult female is 63.8 in tall.

The average adult female weighs 164.7 lb.

If you look at this data from the perspective of Body Mass Index (BMI) you find that on average, both men and women have a BMI of 28.4. It’s no surprise that this number wriggles our plump behinds well into the “overweight” category. According to the US Dept. of Health and Human Services, a healthy BMI is between 18.5 and 24.9. To get our average BMI as a country to a healthy 21.7 (right in the middle of the healthy range), each grown man would have to lose 45.7 lb and each woman 39.2. That’s a lot of extra fat!

To see just how much extra weight we’re carrying in total in the USA, I looked at some Demographic data from the US:

Adults age 20 and over account for 72.7% of the country’s 308,745,538 inhabitants.

There are 0.97 males for every female in the country.

If you crunch the numbers, you find that adults in this nation are carrying 4.32 BILLION KILOGRAMS (9.52 BILLION LBS) of extra fat! But how much fat is that really? Well, after considering that fat has a density of 900 kg/m^3, I can assert that we have this much blubber to spare:

  • Enough fat to cover the state of Rhode Island 1.5cm deep
  • Enough to submerge the island of Manhattan ankle deep  in a layer of sticky biohazardous goo (8 cm)
  • Enough to make a rope 2 cm in diameter that could reach to the moon.
Sadly though, The Simpsons got this one wrong- not just a little wrong, but a few orders of magnitude wrong. The Grand Canyon, with a volume of 4.17 trillion cubic meters, would not be filled 2/5 full. In fact, it would only be filled 0.0001% full. Why, Kent Brockman, why?
9.5 billion pounds you say?




Hello world!

2 06 2011

Hello dudes and dudettes! Welcome to my corner of the internets. It’s going to get weird.

Told you it would get weird.