Due midnight, Wednesday October 2
Collaboration policy for COS 109:
Working together to really understand the material in problem sets and
labs is encouraged, but once you have things figured out, you must part
company and compose your written answers independently. That helps
you to be sure that you understand the material, and it obviates questions
of whether the collaboration was too close.
You must list any other class members with whom you collaborated.
Problem set answers need not be long, merely clear enough that we can understand what you have done, though for computational problems, show enough of your work that we can see where your answer came from. Don't forget that if the data you start with is approximate, the results cannot be precise. Please type your answers if at all possible, especially if your penmanship, like mine, is peccable. Thanks.
This pset, like others, is most easily done if you use scientific notation, with powers of ten like 10^12 and 10^15 instead of words like trillion and quadrillion.
(a) A technology story says that the state of the art in the manufacture of integrated circuits is about 1,000 transistors per square micron. If this is true, about how many transistors would there be in a 1 cm by 1 cm circuit? (A micron is one millionth or 10-6 of a meter.)
(b) A new integrated-circuit factory is going to produce wafers that are 16 inches in diameter. Supposing that there are 500 chips of a particular type on an 8-inch wafer, and assuming that everything else is unchanged, approximately how many chips of that type will there be on a 16-inch wafer?
(c) A new Apple iPhone 16 has a 6.1 inch screen (measured diagonally). An iPhone 11 from a few years ago has the same screen size, but the iPhone 16 has twice as many pixels on its screen than the iPhone 11 does. The resolution of the new phone is 460 pixels per inch. What is the resolution of the iPhone 11, in pixels per inch?
(d) A friend reports being offered two 9-inch pizzas as an alternative to one 18-inch pizza, at the same price. Is this a fair trade, a rip-off by the pizza maker, or a good deal for the pizza eater? Explain your answer quantitatively.
A story on Slashdot a couple of years ago says that "Earth has 20 quadrillion ants". An author of the study says ""We simply cannot imagine 20 quadrillion ants in one pile, for example. It just doesn't work."
(a) Suppose the 20 quadrillion ants were packed tightly into a cube. How long would each side of the cube be, in meters?
(b) The Slashdot story goes on to say "The total mass of ants on Earth weighs in at about 12 megatons." How much does an average ant weigh, in ounces (or grams if you prefer metric)?
[For more detail, see this Hacker News discussion, which also points to the original article.]
"The surging popularity of the Twitter messaging service has broken at least one Twitter client application and affected another as a part of what is being called 'the Twitpocalypse.' Each message on Twitter is assigned a unique identification number. On Friday evening, the number of tweets exceeded 2,147,483,647 and as a result some programs stopped working properly." (Macworld.com, 6/13/09)
(a) Computers often store and process integers using one bit to represent the sign (positive or negative) and the remaining bits to represent the value. If this is the case, what is the likely size of the integers being used by Twitter's computers, in bytes?
(b) If the computer instead uses a representation that assumes all numbers are positive (no sign bit), what would be the number where the value becomes too big? Use an expression; you don't need to compute it exactly.
The "Rule of 72" is a very useful rule of thumb for estimating the effects of compounding, where some quantity grows by a fixed percentage in each of a series of identical time periods. The rule of 72 says that if a quantity is compounding at x percent per time period, it will double in approximately 72/x periods.
For example, if gas prices are rising 8% per year, in 72/8 or 9 years a gallon of gas will cost twice as much as it does today. But if prices are only rising 6% per year, doubling will take 72/6 or 12 years. Conversely, if the doubling time is given, you can compute the number of periods by dividing 72 by the number of periods: if something doubles in 10 years, the rate is 72/10, or about 7% per year. The approximation breaks down if the rate is too high, but it's good enough for typical values. (The web site given above has a good explanation of how it works, and a simple Javascript implementation that you might enjoy looking at. But don't use it to do the calculations below, and don't use your calculator; this problem is about learning to do your own arithmetic in your head or at least with pencil and paper.)
(a) Princeton tuition grows significantly every year, as you've probably noticed. Suppose that 25 years from now, you're trying to send your kids to Old Nassau and you discover that the tuition has quadrupled from what it is now. Approximately what annual rate of tuition increase would that correspond to?
(b) In a Turing Award talk in June 2018, computer architect Dave Patterson said that processor speeds no longer double every two years, as they did 30 or 40 years ago; instead their speeds are increasing only 3% per year. If this is true, in what year will processors be twice as fast as they are now in 2024?
(c) The Newark Star Ledger said on 8/23/22 that "Swiss glacier volume has shrunk by half from 1933 to 2016 (85 years). Since then, glaciers have lost an additional 12% in 6 years." If we simplify by saying that glacier volume shrunk by 12% in the six years from 2016 to 2022, and assume that the rate continues for the future, in what year will it have shrunk to half of what it was in 2016?
(d) In a suprisingly boring book about disease called The Mosquito, the author has one interesting factoid: "In one day a single bacterium can spawn a culture of over four sextillion (21 zeros) bacteria." What is the doubling time of a bacterial culture, that is, how long does it take for the number of bacteria to double?
Please use this Google Doc for your answers. It would be a great help if you could type your answers.
Submit your problem set in PDF format by uploading a file called pset4.pdf to Gradescope. You can submit as many times as you like; we will only look at the last one.
A parting thought about big numbers from an article on numeracy for lawyers by Robert James:
A million seconds ago? You can kind of conceive that; that was last week (11 days). A billion seconds ago, though, most law students had not been born (32 years). And a trillion seconds ago, your ancestor might have been dating a Neanderthal, which may explain some things (32,000 years).