Today we spent the entire period on the homework. I first guessed which coin tosses were real and which were fake. I got 9 right and 5 wrong. We discussed how difficult it is for people to choose random numbers; their notions of randomness don’t agree with “really random”. I asked everyone to think of a number between 1 and 10. I then guessed that lots of you thought of 3 or 7, and indeed about half the class chose one of those. To my surprise, one person chose 1, which is very unusual.

Quickly then, on Problem 1, just use a rectangular-shaped building with a certain base and height. Pretend that the balls are cubes of whatever size. The answers were pretty much in the right ballpark. I remarked that you can ignore the pyramid at the top; it’s so small and the errors of the other numbers are going to swamp it. I also mentioned that it is overkill to keep more than two significant digits in these calculations, for the same reason. Finally, I encourage everyone to use powers of 10 (scientific) notation.

Problem 2: People used pretty much a rectangular shape again, and got reasonable answers. In problems like this, it’s a lot easier to use metric rather than English units because the conversions are easier. I mentioned another way to estimate how long it takes to fill Lake Champlain…the depth to which a given amount of precipitation will fill the lake is equal to the precipitation in the watershed times the ratio of the area of the watershed to the area of the lake. I estimated that to be about 100, and with an average lake depth of 200 meters, and a precipitation in a year of a meter or so, it would take just 2 years to fill the lake. I noted that some had really long times; in one case, that was due to an arithmetic error (do pay attention to that!) Since the lake has only been there since the last ice age and is hence about 20,000 years old at most, any estimate longer than that can’t be right. This is known as a “sanity check.” You should always try to think of such checks. If you can’t figure out what’s wrong, you can at least tell me in the paper that you applied such a sanity check and don’t believe your calculation.

In problem 4, several teams just guessed. One had a number that they remembered from another source. I outlined my method: I know that I can’t see individual cells, but I can see them at moderate magnification (about 100x). So they must be smaller than 1/10 of a millimeter, but not much smaller, maybe about 1/100 millimeter or 0.00001 meters. That corresponds to 10^{15} cells per cubic meter. I also guessed that my body is about 0.1 cubic meters in volume by pretending that I am a box. That gives 10^{14} cells in my body, which turned out to be not that far from the estimate that the one group that used what they remembered got. I also noted that these are just the human cells. Actually about 90% of the cells in our body are not human, but bacteria. Most of them live in our gut, and they perform essential tasks in our bodies. We could not live without these beneficial bacteria.

On problem 5, the estimates were all in the right ballpark. There are about 10,000 students at UVM, and if each of them uses 10 books in their classes (5 classes, 2 books per class) that yields about 100,000 books. Everyone’s estimate was fairly close to that.

On problem 6, again the approaches were similar. If a shopper spends about an hour in the mall (60 minutes), and you command 20 places, that would be 3 minutes per place. A refinement would be to note that you are likely to arrive anytime between when the last place became vacant and the next one comes free. So maybe 1.5 minutes is also a good estimate.

On problem 7, only one person had been to the Grand Canyon, which is much larger than most people realized. It’s probably about 10 km across, 1-2 km deep and 100-200 km long. Some people had the strategy of estimating the population and how much ice cream each person eats in a year; one team estimated the dollar amount of the ice cream made every year and worked from that. The answer is, we don’t eat nearly as much ice cream in a year to fill the Grand Canyon.

On problem 8, estimate the number of households as 300,000,000/3 if you have an average household size of 3. (I pointed out that not everyone is married, and not all married couples have any children at all while others have more than average. But using 3, which is probably not far off, makes the numbers easier). Although one team estimated the proportion having $1,000,000 in net worth at something like 20%, the actual proportion is smaller, just a few percent.

On problem 9, most just estimated the population of Burlington, the proportion of these that had pianos, and how often they needed to be tuned and worked from there. One group had the ingenious idea of assigning one tuner to each of the three music stores in town. All the numbers were “several.” I pointed out that Burlington’s tuners would probably serve a wider area, maybe down to Middlebury and up to Grand Isle. The actual number I got from the phone book a few years ago was about 5.

On problem 10, this is a geometric series, so the sum is equal to 2^{64}-1. Ignore the 1, it is negligible. I then pointed out a useful fact, that 2^{10} is almost equal to 10^{3}. So 2^{60} is pretty close to 10^{18}, and the number of grains would have to be 16 times this, or about 1.6×10^{19}.

I passed out a new problem set, due next Wednesday.

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