Wikipedia:Reference desk/Archives/Mathematics/2008 April 11

= April 11 =

Polar to Cartesian
Given the formula $$\frac{r(1 + e)}{1 + e cos z} \!$$, where r is the radius from the focus to the ellipse border, e is the eccentricity, and z is the angle, how do you reformulate this into Cartesian coordinates?--  十  八  03:39, 11 April 2008 (UTC)
 * Write r and z as functions of x and y. Bo Jacoby (talk) 08:09, 11 April 2008 (UTC).
 * Start by defining the focus to be (x,y)=(0,0). The work out a formula in x and y for the distance from the origin (using Pythagoras). Then work out a formula in x and y for the angle, using basic trig. Then substitute it all in and simplify. --Tango (talk) 11:43, 11 April 2008 (UTC)

GMT-8

Odds
Is there a simple way to calculate the probabilities of an 11-year old boy in a developed country in the year 2008 living to a very old age, say 150? It was asked by him at the miscellaneous desk, but I'm curious about what the math answer might be if there is one. Any answer would help. Thanks, Julia Rossi (talk) 11:59, 11 April 2008 (UTC)


 * I would say probability=$$\lim_{x \to 0}x%$$. C'mon, 150 is almost world record. Have you ever heard of anybody live over 120 in this 10 years? Visit me at Ftbhrygvn (T alk |C ontribs |L og |U serboxes ) 12:51, 11 April 2008 (UTC)


 * Well, you have to consider that those people were born around 1900, healthcare was rather different then. I'd say it's entirely possible, but I don't know quite how you'd calculate it. In fact, I'd say it's impossible to calculate because it relies on knowledge of future conditions - whether global warming causes the spread of tropical diseases, how safe cars are in the future, how well we fend off the effects of ageing... You can't give odds because we can't say what will happen. 150 years is a LONG time. Consider that 150 years ago there were no such things as cars, planes, computers, fridges, plastics, widespread electricity, all the things we take for granted today. -mattbuck (Talk) 13:14, 11 April 2008 (UTC)


 * You can always calculate odds. If we don't know what's going to happen you work with the probabilities of what is going to happen. Probability is all about incomplete knowledge (unless you're working with Quantum Mechanics where things are actually random), so incomplete knowledge isn't going to stop you calculating it. --Tango (talk) 14:27, 11 April 2008 (UTC)


 * It's misleading to speak of "calculating" odds here. The subjectivist Bayesian_probability interpretation would be that any given individual can think about this situation and/or look at available evidence and come up with their own personal probability/odds (essentially, what their own betting odds would be on this). This is a complex situation and different people will likely come up with substantially different odds (as compared with rolling dice or flipping a coin, where almost everyone will come up with the same odds).


 * I'll also note that there are issues (in coming up with your own personal odds) with simply extrapolating from current data, as we do not know what the limits on human lifespan are, nor what medical advances may come in the next 100 years or so. kfgauss (talk) 07:35, 12 April 2008 (UTC)


 * You calculate the chance of someone living to a certain age using life tables, however no life tables are going to go up to 150 years since we have no evidence to base it on. You would have to decide on a way to extrapolate it, but with so little information to go on, it's going to be a pretty meaningless number. --Tango (talk) 14:27, 11 April 2008 (UTC)


 * Tango is right. Any mathematical model would be extrapolating so far beyond known data (the oldest person to have lived so far died at the age of 122) that it would be meaningless. There may be enough data to give a statistically meaningful answer up to an age of 110 or so, but beyond that - no chance ! Gandalf61 (talk) 14:46, 11 April 2008 (UTC)


 * Also note that for extremely old ages the chances of survival decrease probably at least exponentially because the body’s cells start running out of telomeres. At that point healthcare and modern technology become pretty much trivial factors. For a very long life span they’re certainly a limiting factor, and so you could make a model assuming a person lives an ideal life, figure out how many telomeres a person starts with (a random variable), the average rate of mitosis through the various stages of life (another few random variables), and figure out the probability of living to various ages of interest (again assuming an ideal life).
 * Not knowing the specifics, I don’t know what ages this model could be useful at: as it would only be applicable for people who have lived sufficiently old to get past the other hurdles of things that lead to an early death.
 * Another thing, although mostly unrelated, is to note is that a documented case of anyone living over 120 could potentially spark quite a religious debate because in the Bible, God is quoted as saying that man is limited to 120 years. GromXXVII (talk) 22:20, 11 April 2008 (UTC)


 * Um... Jeanne Calment... --Tango (talk) 22:40, 11 April 2008 (UTC)


 * Well, he did say that man is limited to 120 years. Black Carrot (talk) 23:18, 11 April 2008 (UTC)


 * Hi. I don't think this is a reliable source, but I think on a program called "Natural Cures" or something, it said that a man that is historicly documented lived to 150 years. Also, if I remember correctly, the Bible mentions Adam and Eve living to about 1000 years, and Noah living to about 700 years, but I don't remember exactly. Also, I don't really see how you can calculate such odds, if you don't know the exact circumstances of such future or the odds of such, or how you can calculate the possibility without a certain documentation of such an age, or the googolplexes of possible factors affecting this unlikely outcome. Thanks. ~ A H  1 (TCU) 23:55, 11 April 2008 (UTC)


 * Does this mean then, with things like telomeres to take into account, even given the history of life-expectancy increments, plus genetic intervention, changes such as ideal non-polluted environment, the body would have to stop around 120? Julia Rossi (talk) 01:12, 12 April 2008 (UTC)


 * My 2 cents, I would doubt that anyone born yet or in the next 20 years will reach 150, our environment is simply too polluted for that to happen I think. I'm not convinced that humans are limited to around 120 years maximum life span either, I'm no expert I doubt one can conclusively show that the rate of loss of Telomeres is necessary even approximately fixed either and I suspect is very much related to environmental conditions. I also doubt our environmental conditions will improve anytime soon either. As for the math, if you make a ton of assumptions (which removes it's relationship the real world which I assume you don't want) then yes it's calculable. A math-wiki (talk) 02:43, 12 April 2008 (UTC)


 * Well, leaving the real world aside, what would be the calculation process in math (and some english)? Would it involve a life mean (with projections taken into account), something like Ftbhrygvn's above with a couple of explanations? Nothing too involved, this is my first time here, thanks Julia Rossi (talk) 03:42, 12 April 2008 (UTC)


 * To clarify what I said a little, the telomeres comment and the age 120 have nothing to do with each other (I just thought the latter was interesting and noteworthy, while the former could actually be used to make a model). Every time a cell reproduces a telomere is expended, and so in a sense that starting value is determined at conception, although varies to some extent, but I don’t know the distribution.
 * As for when it becomes a factor that could lead to death, that also depends on a large part on how a person lives their life: because the bodies cells will reproduce when they need to, to account for losses for whatever reason. For instance, skin and bone marrow should be some of the first parts of the body to encounter this kind of problem because of it’s consistent reproduction throughout life.
 * What kind of lifestyle would be the needed ideal for this model? I’m not quite sure, but probably something fairly well balanced. For instance an athlete’s extreme exercise will cause greater cell reproduction, and a couch potato’s lethargy could cause perfectly good cells to go bad and cause others to reproduce to take its place.
 * That said, the ability to replace organs or large parts of the body could nullify this cause, especially because the brain is [I think] one of the lease active parts of the body in cell division, that by the time the brain runs out of telomeres, we’re talking about a timescale of probably tens of thousands of years if not more. However, there would be huge percentage of the body to replace to get around the decaying telomeres: it would probably be easier to figure out a way to graft telomerase (enzyme that adds telomeres) into the first somatic cells of a new organism.


 * As for an actual model for this, I’ll assume an ideal lifestyle, and a lucky enough person to avoid all disease or accidental death. Take one vital organ, say the skin, and some average starting number of telomeres S. While the body is capable, assume the skin has a roughly constant rate of use through adulthood, say R. That gives roughly S/R years until they run out and their skin can no longer function and death eventually follows. This has some obvious error because the rate will be greater during childhood growth, and less during old age when one starts to run out of telomeres. These errors have opposite signs, and so to some extent will cancel each other out: so perhaps with some luck S/R will be accurate to within a decade or two. There could be other errors though, like for instance I’m just guessing that skin has a constant rate of growth through adulthood.


 * Now if you want a real model, I’d say to find different models based on the leading causes of death at various age brackets, and weight them accordingly. For instance, childhood death is usually caused by something else, so take a model for that and weight to be the predominant influence during those years. Likewise the first few decades of old age will probably be more heavily affect by other things than telomeres. The error analysis of such a model will be quite ugly though. GromXXVII (talk) 12:01, 12 April 2008 (UTC)


 * The "natural limits" may no longer apply in another century. By that time we will be able to replace any body part that malfunctions, with the exception of the brain, because, of course, if your brain is replaced entirely it isn't really you any more.  However, there may be ways we can replace it a few cells at a time and thus end up with the same "pattern", even though all the individual cells have been replaced.  Now, as to how to calculate odds, I would take mortality tables from 1900, 1910, 1920, etc., and extrapolate how life expectancies increase with each decade.  This wouldn't be all that accurate of a method, as it would be extrapolating well beyond our current data, and the more you extrapolate, the more the actual value tends to vary from the expected value. StuRat (talk) 04:00, 12 April 2008 (UTC)


 * For your fantastic help and consideration, my thanks to all, Julia Rossi (talk) 08:34, 12 April 2008 (UTC)


 * "it said that a man that is historicly documented lived to 150 years." Sounds to me like a biblical story. Imagine Reason (talk) 01:57, 13 April 2008 (UTC)

Rational Exponent solver
Is there a website that will solve any rational exponents? —Preceding unsigned comment added by 192.30.202.21 (talk) 19:56, 11 April 2008 (UTC)
 * What do you mean by a "rational exponent"? --Tango (talk) 21:04, 11 April 2008 (UTC)
 * Hi. I'm just guessing, and I'm not an expert on this, but perhaps an exponent by a rational number, like ^2.7 or ^7.9797..., and not an irrational number like ^π or ^$$e$$, and would a calculator work, or do you need an exact answer, as calculators can usually only hold finite number series? Thanks. ~ A H  1 (TCU) 00:02, 12 April 2008 (UTC)
 * Well, yes, that's what the words would normally mean, however the OP must mean something else, since you can't "solve" an operation. --Tango (talk) 00:05, 12 April 2008 (UTC)
 * Perhaps, in the idiolect of the questioner, "to solve" means "to compute the value of something involving". --Lambiam 01:59, 12 April 2008 (UTC)
 * In which case, any scientific calculator will do the job. --Tango (talk) 13:24, 12 April 2008 (UTC)
 * Google will tell you that 3.14^2.72 = 22.4723579. --Lambiam 02:01, 12 April 2008 (UTC)

Race
You have 25 horses, and a track on which you can race five of them at a time. You can determine in what order the horses in a race finished, but not how long they took, and so can not compare times from one race to another. A given horse runs at the same speed under all circumstances, and no two horses run at the same speed. How many races does it take to find the three fastest? I've found a way to do it in 7, and it can't be done in 5, so 6 is the big question mark. Black Carrot (talk) 23:14, 11 April 2008 (UTC)


 * I can do it in 25C5 races. I choose not to try and optimise my solution. -mattbuck (Talk) 23:21, 11 April 2008 (UTC)


 * Each horse would have to compete in 10626 races. I'm pretty sure there are laws against that. Black Carrot (talk) 23:32, 11 April 2008 (UTC)


 * He's my method, first off race 5 of them, and then take the 3rd horse and race it against 4 new ones, if anyone beats the 3rd horse, then the second horse will be in the next race as well, to determine if the new horse is 3rd, or 2nd or better. I'm also to lazy to check the efficiency of my method. A math-wiki (talk) 02:50, 12 April 2008 (UTC)


 * I don't think it fares too well if you pick slow horses near the beginning, since you'd focus a lot on getting rid of them one at a time. Black Carrot (talk) 03:20, 12 April 2008 (UTC)


 * What method do use to get 7 races ? StuRat (talk) 03:45, 12 April 2008 (UTC)


 * I think I've figured out the 7 race method. First you run each group of 5 in a race, for 5 races total.  Then, in the 6th race, you race the champions, to get rankings for all 5 of those.  Below are the results, with A1-E1 representing the champions, from fastest to slowest, and the remaining horses listed behind the champions in the order they finished in the eary races:

A1 A2 A3 A4 A5 <- early race order B1 B2 B3 B4 B5 <- early race order C1 C2 C3 C4 C5 <- early race order D1 D2 D3 D4 D5 <- early race order E1 E2 E3 E4 E5 <- early race order ^ | Champions order


 * Now, we know the fastest horse is A1. The three fastest horses, in order, could be A1,A2,A3 or A1,A2,B1 or A1,B1,B2 or A1,B1,C1.  Race A2,A3,B1,B2, and C1 to determine the 2nd and 3rd fastest horses. StuRat (talk) 04:27, 12 April 2008 (UTC)


 * That's exactly what I got. I wouldn't be surprised if it was unique, though I wouldn't know how to prove it. What do you think about the six-race case? I noticed that if you already know the three best, it's possible to prove you're right in only six races, so it's a close call either way. Black Carrot (talk) 07:14, 12 April 2008 (UTC)


 * I can't see any way to do it in 6 races. StuRat (talk) 13:58, 12 April 2008 (UTC)


 * I suppose so, but in many cases it could be quiet efficient. A math-wiki (talk) 19:56, 12 April 2008 (UTC)


 * I don't think it affects your answer, StuRat, but you left out a possible order: A1,B1,A2. --Tardis (talk) 15:15, 14 April 2008 (UTC)


 * Yes, good catch. StuRat (talk) 18:43, 14 April 2008 (UTC)