Wikipedia:Reference desk/Archives/Mathematics/2009 May 22

= May 22 =

HOW HE DID IT?
On a web I come across a problem that runs as follows: (http://www.cimt.plymouth.ac.uk/resources/help/h1alge1.pdf )
 * "Han Sin, a Chinese general, devised a method to count the number of soldiers that he had. First, he ordered his soldiers to form groups or 3, followed by groups of 5 and then groups of 7. In each case he noted down the remainder.  Using the three remainders, he was able to calculate the exact number of soldiers he had without doing the actual counting."

Please help me to know HOW HE DID IT? Kasiraoj (talk) 08:01, 22 May 2009 (UTC)


 * Possibly a chinese remainder theorem has something to do here...? --CiaPan (talk) 08:07, 22 May 2009 (UTC)


 * Hardly a general if he only did those numbers, sounds like the equivalent of a lieutenant or possibly a major with a small company. Unless the Chinese had rather inflated ranks then. I suppose it's possible he was in something like intelligence where you don't need so many people below you to get high rank, it would explain also why he counted in such a strange way that gave little information away as it was being done. Perhaps it was an early form of this Zero knowledge average, the wiki article Zero-knowledge protocol doesn't seem to have much on this sort of thing. Dmcq (talk) 10:31, 22 May 2009 (UTC)
 * Well, all solutions would be congruent modulo 105, so the army must be small enough that a rough estimate is accurate to the hundreds. This should be achievable with sizes within an order of magnitude of 1000.  So General could be correct. Taemyr (talk) 23:39, 23 May 2009 (UTC)


 * I'd have thought a general would command a lot more. It's hard to find from wiki but I know of a colonel who was in charge of a few thousand troops and a general is a few ranks above that. I'm surprised he got so far, he refused staff officer training the first time because his wife wanted him at home. Dmcq (talk) 13:24, 24 May 2009 (UTC)
 * The lowest rank of general (a Brigadier or Brigadier General) typically commands a brigade (hence the name) of a few thousand men. Sometimes they are commanded by colonels, too, though, so that's probably what the colonel you knew commanded. --Tango (talk) 15:15, 24 May 2009 (UTC)


 * Hmm, I was going to suggest looking at our article about Garner's algorithm, but we don't have one. Maybe someone can fix that.  It should be in Knuth vol. 2 and places like that.  Anyway, yes, the story is obviously supposed to be an illustration of the Chinese remainder theorem. 207.241.239.70 (talk) 10:35, 22 May 2009 (UTC)

probability book?
Anyone got a favorite introductory probability book for math nerds? I'd like something mathematically solid (free of hand-waving; e.g. it should introduce sigma-algebras at the appropriate place) but basically application-oriented rather than foundational, discussing common distributions and how to calculate stuff with them, etc. It would be good to also have some coverage of basic statistics, stochastic processes like Brownian motion, and maybe some trendy subjects like machine learning. Thanks. 207.241.239.70 (talk) 10:32, 22 May 2009 (UTC)


 * My favorite is Standard Probability and Statistics Tables and Formulae by Zwillinger and Kokoska. Warning: The book is less a text than a "handbook" -- it assumes that the reader has a solid foundation in mathematics. Wikiant (talk) 10:42, 22 May 2009 (UTC)
 * Thanks, I may look for that, but I think I want a textbook with exercises. I'm starting to feel slightly clueful about the subject through surfing wikipedia, but that's usually a bogus feeling, since the understanding hasn't really sunk in. 207.241.239.70 (talk) 11:54, 22 May 2009 (UTC)


 * What about this one by Grinstead & Snell, available for free download from the website of Dartmouth College? --NorwegianBluetalk 13:51, 22 May 2009 (UTC)
 * Nice!!!! Online books are the greatest.  This one is a little more elementary than I had in mind, but it's probably a good way to get started anyway.  Thanks! 207.241.239.70 (talk) 14:29, 22 May 2009 (UTC)

Maximum Volume of Cone made from cut sector of a circle.
Let's say we have a circle with a radius R cm. A sector can be cut out to form a cone,by joining the straight sides of the sector together. What is the angle of the sector to be cut out such that the maximum of volume of the cone can be achieved? I have already arrived a few formulas of the cone. Radius of cone's base= $$\pi R\theta/180$$ height of cone =$$\sqrt{R^2-(\pi R \theta/180)^2}$$

is differentiation needed? if yes, can you please explain in more detail as i have not learnt it yet. —Preceding unsigned comment added by Invisiblebug590 (talk • contribs) 11:43, 22 May 2009 (UTC)


 * It's not clear to me that you need to use calculus for this, but yes, differentiation is a standard way to solve maxima and minima problems. You've got formulas for the cone's base and height.  So you should be able to find a formula for the volume as a function of theta.  Notice that if theta is very small, the cone is thin and needle-like with small volume.  And if theta is very large, the cone is flat like a slightly pointed pancake; again the volume is small.  If you draw a graph between those, there is a maximum somewhere, and the derivative is zero at the maximum.  So you have to differentiate the function and find where the derivative is equal to zero.  207.241.239.70 (talk) 11:51, 22 May 2009 (UTC)


 * I'd rather declare some symbol for the cone's base radius, say r, and write
 * $$r = R \tfrac \theta {2 \pi}$$
 * where &theta; is the sector's angle in radians. Then the cone's height
 * $$h = \sqrt{R^2 - r^2}$$
 * so its volume
 * $$V = \tfrac 13 \pi r^2 \cdot h = \tfrac \pi 3 r^2\sqrt{R^2 - r^2}$$
 * The volume approaches zero as r becomes zero (because the $$r^2$$ term becomes zero) or when r approaches R – these are two cases described by 207.241.239.70 above. Between these limits volume is non-zero, positive continuous, and it reaches a maximum somewhere. We may further simplify expressions if we note, that the volume V is maximum iff its square V2 is maximum. So we write:
 * $$V^2 = \mathrm{some\ constant}\cdot r^4\cdot(R^2 - r^2) = \mathrm{some\ constant}\cdot (R^2r^4 - r^6)$$
 * If we consider r growing from 0 to R, the difference in parens grows as long as the growing speed of $$R^2r^4$$ is greater then that of $$r^6$$. When the two become equal the expression gets its maximum value and falls back as $$r^6$$ grows faster than $$R^2r^4$$.
 * A 'growing speed' is simply a derivative, which can be easily calculated for the expression above – see rules at derivative and List of differentiation identities.
 * The equation $$(V^2)^\prime = 0$$ transforms to $$(R^2r^4)^\prime = (r^6)^\prime$$
 * When you substitute appropriate expressions for derivatives you'll get a simple equation with r. Solve it, and then calculate &theta; from the first equation above. --CiaPan (talk) 14:41, 22 May 2009 (UTC)


 * I calculate that the angle to be cut out for greatest conic volume is about 66 degrees. An interesting (to me, anyway) extension is to use the removed part of the original disc to make a second cone, the objective now being to maximise the combined conic volume. The expression to be maximised is rather more complicated than before, so it's easiest to get an answer numerically. It's obvious that there'll be two maxima, as either portion can be deemed to be cut out or left behind. What's maybe surprising is how flat the volume function is, with the value within about 1% of the maximum for all angles between 90 and 270 degrees.→86.164.72.176 (talk) 18:06, 23 May 2009 (UTC)

working out foot pound ratio for powder in grams
Hi

Im working on a experiment involving a small cannon, and need to know foot/ pound ratio for grams of gunpowder. I learned from a documentary, that 660 pounds can launch 2700 pounds 23 miles. How far could 30 grams of powder move a 165 pound projectile? i know its only a matter of 4-6 feet, but I would like to know exactly. can someone help me with these numbers? Rob —Preceding unsigned comment added by 79.68.219.232 (talk) 17:53, 22 May 2009 (UTC)
 * You want to figure out the kinetic energy that it takes to heave the 2700 pound shell 23 miles. Assume that's the maximum range, which means the gun is fired at a 45 degree angle, so the shell exits with equal horizontal and vertical velocity components.  Is that enough for you to finish the calculation?  Keep in mind that the big cannon may convert its gunpowder energy to the shell much more efficiently than your small cannon.  207.241.239.70 (talk) 20:35, 22 May 2009 (UTC)


 * I imagine that real-world issues like air resistance and cannon "efficiency" (proportion of energy that goes into kinetic energy versus heat, noise, recoil, friction, etc.) would make any such calculation without extensive experimental work hopelessly incorrect. At high speeds, at least, I bet that neglecting air resistance would cause you to be wrong by a factor of two or more.  And efficiency probably varies wildly from one cannon to another, as 207.x.x.x said.  The only feasible way I see to get a good estimate is to take your cannon and carefully experiment with different payload masses and different powder quantities and try to work out a reasonable estimator.  (Also, your type of gunpowder might be different from the gunpowder mentioned in the documentary.)  Eric.  131.215.159.91 (talk) 21:36, 22 May 2009 (UTC)


 * For example, assuming 100% efficiency and no air resistance and some other unreasonable approximations tells me that 660 pounds of explosive can send 2700 pounds of payload a distance of 239.5 km (148.8 miles), which is much larger than the actual 23 miles. Eric.  131.215.159.91 (talk) 21:48, 22 May 2009 (UTC)