## Posts tagged ‘population’

### Where Would We Fit?

In the classic book *101 Puzzle Problems*, Nathaniel B. Bates and Sanderson M. Smith make an astounding claim:

*The volume of the 1970 world population is less than the volume of the Houston Astrodome.*

That is, if you packed all of the people on Earth who were alive in 1970 — all 4 billion of them — as tightly as possible, they would’ve actually fit in the stadium, and they could’ve seen Evel Knievel jump 13 cars or Billy Jean King defeat Bobby Riggs.

Before you read any farther, you might wish to do a Fermi estimate to verify the reasonableness of this claim.

It’s an amazing fact!

Unfortunately, it’s also wrong.

Bates and Smith argue that the average human body has a volume of approximately 2 cubic feet. Their argument? That water weighs 62.5 pounds per cubic foot, and “the human body consists mostly of water.” Fair enough, if we think that an average human weighs about 125 pounds, which they probably didn’t, even in the 1970s.) Personally, I would’ve attacked the estimate a little differently. An average human can fit in a box that is 6 feet tall with a base that measures approximately 0.5 square feet — I know; I’ve been to funerals — and such a box has a volume of 3 cubic feet. But there’s a fair amount of wasted space in that box so, yeah, 2 cubic feet seems like a reasonable estimate.

Therefore, the world population in 1970 would have had a combined volume of 8,000,000,000 cubic feet.

So far, so good.

But Bates and Smith then make their statement about the Astrodome, with no calculations or estimates about the volume of the stadium to justify the claim. Please understand, I don’t fault Bates and Smith; as I understand it, the claim about everyone fitting in the Astrodome was a ubiquitous factoid during the decade of bellbottoms and disco. Still, you’d think that two mathematicians would have checked the stat before including it in a book.

The Astrodome has an outside diameter of 710 feet and a height of 208 feet. A cylinder with those dimensions would have a volume of

*V* = π × 355^{2} × 208 ≈ 82,000,000 cubic feet

Although this is surely an overestimate, it’s still an order of magnitude too small to fit the world’s population in 1970. There’s not nearly enough space to store all those buggers.

Bates and Smith state that the volume of the 1970 population “takes up about one-eighteenth of a cubic mile.” And I think therein lies the error. The height of the Astrodome is just shy of one-eighteenth of a mile, and the diameter is a whole lot more than one-eighteenth of a mile, so someone (incorrectly) assumed that a container with those dimensions would easily hold 1/18 cubic mile. But that’s not right. As any geometry student will tell you, shrinking each dimension to a fraction of its original length results in a volume that is the *cube* of that fraction. Oops.

The diameter of the Astrodome is about 1/7 of a mile, and the height is about 1/22 of a mile, so its volume would have been less than 1/1000 of a cubic mile.

So, what container would have been big enough to hold the 1970 population?

Honestly, who cares?

Things have changed. Today, we are bumping up against 8 billion people worldwide, the Astrodome was declared “unfit for occupancy” in 2016, and instead of calculating the volume of stadia, we can simply use Google to find the volume of some really large places.

In fact, Google claims that the volume of the Astrodome is 42,000,000 cubic feet. If we can believe what we read, then the calculations above were, sadly, unnecessary.

So, you may be wondering, **what places in the world could hold all of us**? That is, what has a volume of 16,000,000,000 cubic feet?

The building with the greatest volume in the world is the Boeing Everett Factory. It has a volume of almost 500,000,000 cubic feet — which means you’d need 32 of them to fit today’s world population.

The manmade structure with the greatest volume is the Great Wall of China, tipping the scales at just under 20,000,000,000 cubic feet. That would be large enough, though how would you get all of those people inside? For easier packing, you could turn to nature and just dump everyone into Sydney Harbour, which has a volume of 562,000 megaliters, or about 19,500,000,000 cubic feet.

### How Many Share Your Birthday?

This afternoon, we celebrated Alex and Eli’s sixth birthday with a Disney-themed Cinco de Mayo party. The kids all wore Mickey Mouse ears, while the parents drank lots of margaritas. Tonight’s “bedtime math” question for my sons was the following:

You celebrated your birthday on May 2. How many other people in the world do you think celebrated their birthday on May 2?

It’s a simple estimation problem for most of us, but ratio is a tough concept for six-year-olds. I wasn’t sure they’d make much progress… especially since the good folks at about.com make this claim:

You currently share your birthday with about 859,178 people who reside in the United States.

This estimate appears to have used 313,600,000 as the U.S. population, which is reasonable, and then divided by 365. My frustration is that they then display the result to six significant figures. That’s problematic for two reasons — first, because their population estimate has only four significant figures, but also because it’s not the case that exactly 1/365 of the population celebrates their birthday on a given day.

But I digress. Sure, I’m frustrated with about.com’s negligence, but I started this post to tell you about our bedtime math problem, and it highlighted why I hate traditional textbook problems even more than I hate bad math in the media.

Alex first suggested that maybe the number of people who have the same birthday could be found by calculating 1/14 of 7 billion. When I asked why he wanted to divide by 14, his response was, “Because it’s a multiple of 7.” When I asked a few more questions to probe his thinking, he changed his mind. “No, wait, maybe it’s 1/35.” This time, he said he wanted to divide by 35 because it was a multiple of 7 *and* a multiple of 5, and he knew that 7 billion was also a multiple of both 7 and 5.

Then it hit me. He wasn’t trying to solve the problem. He was just trying to make sure the answer was a “nice number,” that is, an integer that preferably would end in a couple of zeroes.

A few more questions, and he finally admitted he knew an estimate could be found by dividing 7 billion by 365. “But that doesn’t work when you divide,” he told me.

Arrgh.

I believe this is what happens when kids see too many traditional textbook problems where the answers are neat and clean. They get conditioned to thinking that math is never messy.

**[Update: 5/8/13]** Just read this on the About page at the Let’s Play Math blog and thought it was worth including here: “Math is like ice cream, with more flavors than you can imagine — and if all your children ever do is textbook math, that’s like feeding them broccoli-flavored ice cream.”

And that couldn’t be further from the truth. Math is unbelievably messy. At least, real math is. Solving real-world problems often means getting a little dirty. You’ll have to roll around in fractions, dig through some decimals, and — Heaven help us! — occasionally tangle with some irrational numbers and extraneous results.

Eli then offered, “If you divide 7 billion by 365, you won’t get an integer.” (He smiled, proud of himself for using the term *integer*.) “That’s the answer, but I don’t know how to do that.” What he meant is that he couldn’t compute the result in his head; nor would I expect him to. We then found an estimate by building on Alex’s idea — instead of dividing by 35, we divided by 350 to approximate the number of people who celebrated a birthday on May 2, since 350 is close to 365 but gives a much nicer answer.

Wow. There are roughly 20 million people who will celebrate their birthday on the same date as you. Crazy, huh?

All of this reminds me of a few jokes.

Recent research shows that those who celebrate more birthdays live longer.

And all the time, I tell my wife:

Honey, you’re one in a million. Which means that there are 7,000 people on Earth exactly like you, so just remember that it wouldn’t be that hard to replace you.

### Conversion Perversion

“Be there in a jiffy.”

If someone says that to you, then you know that that person should arrive soon. But did you know that *jiffy* is a technical term? Similarly, the expression “two shakes of a lamb’s tail” used to indicate a short period of time, but the unit of time known as a *shake* now has a specific designation.

- 1 sec = 100 jiffies = 100,000,000 shakes

I’m big into conversions. I often tell folks, if you need to convert between televangelists and expatriate poets, the following picture may be helpful to you:

That is, 1 Ezra Pound ≈ 454 Billy Grahams.

The following are some other fun conversions.

- π sec ≈ 1 nanocentury

It’s interesting that this is so accurate. It is within 0.5%.

- 1 furlong per fortnight (FPF) ≈ 1 cm/min

This one is even better. The error is less than 0.000025%.

- 1 m/s = 1 Hz/dpt (Hertz/dioptre)

This is what can happen when common units are replaced with uncommon units. *Hertz per dioptre* is an inside joke among physicists and yet another reason not to hang out with them. (*Dioptre* is a unit of measure for the optical power of a lens.)

- 1 square = 100 square feet

The term *square* is used in the construction industry, typically to measure a roof. For example, if a roof has an area of 1,000 square feet, then the contractor would order 10 squares of shingles. But you wouldn’t want to use this unit in regular conversation, because it leads to awkward phrases like a “one-square square,” which would be a square that measures 10 feet on a side.

- 1 gal ≈ 3 + π/4 L

This is one of my favorite conversions. It’s accurate to 0.00000003%.

- 1 Hubble-barn ≈ 13.1 L

A *Hubble length* is the length of the observable universe (a very, very big length), and a *barn* is 100 square femtometers (a very, very small area), so it’s neat that their product gives a very tangible volumetric result.

- 1 stone = 14 pounds

When asked for my weight, I usually respond, “About 13 stones.” Such a reply leaves room for interpretation, and it could be assumed that I weigh as little as 175 pounds or as much as 189 pounds. And I’m fine with that. What kind of rude bugger asks your weight, anyway?

On a related note, the following formula can be used to approximate the U.S. population for a given year. Let *x* = the last two digits of the year, and let *y* = the projected U.S. population for that year (in millions). Then,

*y*= π*x*+ 276

This result is based on projections from the Pew Research Center. This formula provides an accurate estimate (within 1%) of the actual population for every year since 2000, and it should give a reasonable projection for the next several decades, assuming there are no major catastrophes.

### Qatar, Afar

As the Online Projects Manager for NCTM, I have the privilege of managing the Illuminations project. Illuminations was recently selected as a finalist for a World Innovation Summit for Education (WISE) Award from the Qatar Foundation. Consequently, I have been invited to attend the WISE Summit in Doha, Qatar, from December 7-9, all expenses paid. While there, I will have the pleasuse of meeting Her Highness Sheikha Mozah bint Nasser Al Missned and interacting with education dignitaries from around the globe. I’m very excited to experience the “the unforced fusion of modernity and traditionalism” (http://www.ccnpic.com) in Doha.

**30 Finalists for the WISE Award **

To prepare for my trip, I did some research about Qatar. Through the power of the web, I located a joke that was described as “a very classic Qatari joke.”

The Emir of Qatar was visiting with the president of China.

The Emir asks, “What’s the population of China?”

The Chinese president responds, “One billion. What about Qatar?”

The Emir says, “The population of Qatar is only 250,000 people.”

The Chinese president asks, “And in which hotel are they staying?”

Interestingly, there are more than 1,000,000 residents in Qatar, but nearly ¾ are expatriates living and working in Doha. The ratio of men to women is 3:1, largely explained by the predominantly male expatriate workforce.

I’ve heard that the Qatari people have a sense of humor and a love of mathematics. Consequently, I’ve had a sign made:

طرائف الرياضيات 4 الاهالي ماثي

40 ريال

Translation: “*Math Jokes 4 Mathy Folks*, 40 rials!”

(I sure hope Google translated that correctly!)

### Political Trivia and Humor

I stumbled across the maps of the problematic blog last week, which claimed the U.S. Senate was no longer necessary. It was part of a list of eleven unnecessary things, actually. The list also included phone books, beepers, the Electoral College, and pocket calculators. The author claimed that the Senate gives just 18% of the U.S. population the power to stop a bill from passing Congress. That is, if 50 Senators vote “no” to a bill, then it fails, and the 25 least populous states represent just 18% of the population. I didn’t check the author’s math, but I’ve heard similar estimates before, so 18% sounds reasonable to me.

Interestingly, the author implied that the Senate might have been necessary when it was first created, to give a voice to smaller states. This made me wonder — when Congress first began enacting law in 1789, what percent of the U.S. population had the power to stop a bill?

The first Congress had only 24 Senators from 12 states. (Rhode Island originally rejected the Constitution in 1788, delaying ratification until May 1790, when the federal government threatened to treat them as a foreign government.) Consequently, the Senators from six states had the power to stop a bill.

The populations (in thousands) of the 12 states with Senators in 1789:

- Conecticut: 237
- Delaware: 59
- Georgia: 82
- Maryland: 96
- Massachusetts: 379
- New Hampshire: 142
- New Jersey: 184
- New York: 340
- North Carolina: 393
- Pennsylvania: 434
- South Carolina: 249
- Virginia: 747

The total population (in thousands) of those 12 states was 3,342. The total population of the six least populous states was 568. That means that 568/3,342 ≈ 17% of the U.S. population could have stopped a bill in 1789.

Please understand, I’m not arguing that the Senate should be retained or abolished. But by the numbers, it appears that the Senate might have been even less necessary in 1789 than it is today.

Anyway, here’s a joke about math and politics:

A cannibal goes to the butcher shop and notices that mathematician brain is selling for $1 a pound, but politician brain is selling for $4 a pound. “Is the politician brain really that much better?” she asks the butcher.

“Not really,” he says. “But it takes a whole lot more politicians to make a pound.”