Due 5:00 PM, Wednesday November 11 (after break). Answers need not be long, merely clear so we can understand what you have done. Please submit typed material, not hand-written, if at all possible, and keep a copy for yourself just in case something goes astray. Thanks.
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A reminder about the
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.
Please list any other class members with whom you collaborated. |
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 rate: 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 plenty good enough for typical values. (The web site given above has a good explanation of how it works, and a simple Javascript implementation. But don't use it to do the calculations below; this problem is about learning to do your own arithmetic in your head.)
(a) Moore's Law is often expressed as "capacity doubles every year and a half." What is Moore's Law expressed as a percentage growth per month?
(b) Suppose that about 25 years from now, you're trying to send your kids to Princeton and you discover that the tuition has quadrupled from what it is now. Approximately what annual rate of tuition increase would that correspond to? (Warning to prospective Princeton parents: a recent newspaper story says that college tuition is rising at 6% or even faster at many schools. Start saving.)
(c) Former Princeton professor Ben Bernanke, now chairman of the Federal Reserve, said he wants to keep inflation between 1.5% and 2.5% per year. Assuming he manages to hit the average of these two numbers, when will inflation have doubled the dollar cost of everything?
(d) From TIAA/CREF Participant, 8/03: "In his will, Franklin left 1000 pounds sterling to the cities of Philadelphia and Boston, with the stipulation that the funds be lent out at 5 percent interest a year. Because of compounding, Franklin figured that in 100 years his bequests to these cities would be worth 131,000 pounds [each]." Assuming that 5% was achieved, was his estimate much too high, much too low, or about right, and why do you say so? (The Franklin Institute in Philadelphia owes its existence to his bequest.)
(e) A NY Times story (11/7/07) says that total power consumption by Internet servers is doubling about every 6 years. Suppose that Internet servers used 10 gigawatts in 2001 and 20 gigawatts in 2007. How fast is power consumption growing per month?
(f) Suppose (improbably) that this power consumption continues to double at the same rate. In what year will servers consume 10 terawatts?
(g) An NPR story (10/3/08) states that getting 20% interest for five years doubles your money. Is five years too short, too long, or about right?
(a) "π seconds is a nanocentury." Too high, too low, or just about right, and why?
(b) "Sound travels about one foot in a millisecond." The speed of sound at sea level is about 340 meters/second. Is the quoted speed too high, too low, or just about right, and why?
(c) I was at an over-crowded cocktail party recently, and it started me wondering... Suppose that 10,000 people come to the same party. If they arranged themselves in a square (to make the arithmetic easier), what would be the approximate dimensions of the square?
(d) Approximately what fraction of the football field at the stadium would they occupy? (Note for non-football types: the field is 100 yards long and 50 yards wide; feel free to go metric -- 100 x 50 meters -- if you prefer.)
(e) Suppose 100 million people showed up at our hypothetical party. What would be the dimensions of this gathering?
(f) How does the width of the square depend on N, the number of people in it?
(g) "Video files [...] are typically thousands of time larger than text files. A minute of high-definition video is about 50 megabytes, about 10,000 times larger than an 800-word e-mail message." (NY Times, 9/18/08.) From these facts, estimate the number of letters in the average word in an email message.
(h) A story on Slashdot on 10/22/09 describes "a new fingernail-size
chip that can hold 1 trillion bytes (a terabyte) of data", which is
claimed to be "up to 20 high-definition DVDs or 250 million pages of
text."
(i) Assuming these numbers are correct, what is the approximate
capacity of a high-definition DVD?
(ii) If the books in question are just ASCII text, is the
page-count estimate much too high, much too low, or about right? (As
some kind of cross-check, Pride and Prejudice is 97,680 words,
and Moby Dick is 212,441.)