Wikipedia:Reference desk/Archives/Mathematics/2009 January 27

= January 27 =

Rotating a Right Traingle 180 Degrees
I was trying to fiugre how do I rotate a right triangle with the base facing to the right 180 degrees from the third quadrant to the fourth quadrant. The points of this triangle are A=-2,-4, B=-6,-4, and C=-2,-7. I thought I would make a mistake picturing the rotation in my head. Does anyone know of a simpler method?

Best Reagrds,

--Writer Cartoonist (talk) 02:55, 27 January 2009 (UTC)


 * Draw it on graph paper, and it will be fairly obvious how to rotate it. StuRat (talk) 03:00, 27 January 2009 (UTC)


 * How can a rotation of a half turn move anything from the third quadrant into the fourth? --Tardis (talk) 18:38, 27 January 2009 (UTC)


 * Surely any half turn about a point in the lower half plane moves part of the third quadrant into the fourth. Matthew Auger (talk) 20:00, 28 January 2009 (UTC)
 * As no center for the rotation was given, I was assuming it to be the origin. Maybe the problem is to identify a center that does move it (entirely) into the fourth quadrant: that would be much more interesting!  (The overdone answer $$(10^{100},-10^{100})$$ is easy, though.)  --Tardis (talk) 16:45, 29 January 2009 (UTC)
 * Rotating 180 degrees about the origin just amounts to multiplying all the coordinates by &minus;1. Michael Hardy (talk) 16:52, 29 January 2009 (UTC)

Weierstrass-Enneper parametrization
Hey,

Could anyone give me a hand rewriting the article about the Weierstrass–Enneper parameterization? I've added some information at this page, but I haven't explained quite a bit, hence why I haven't moved it over yet. It would be especially nice if someone could give explanations about why we can write the composite Gauss map as I did and how minimal surfaces and harmonic functions are related (as I have already described the relation between the Gauss map and holomorphic functions, there's not much left to do. I could do with some reference of the statement that G is conformal if and only if M is minimal.). In fact, something should be written about harmonic functions in the minimal surface article...

I mainly based what I wrote on the article here (PDF), so definitely have a look at that. I'm not sure what the author does with harmonic functions and his map J which rotates the tangent space by 90°, so I ended up leaving that out.

It would also be really nice if someone could provide explicit parametrizations for Costa's surface and Riemann's minimal surface (described in the article I linked). I think the articles about minimal surface are really underdeveloped at the moment, and even worse for surfaces of constant mean curvature. I went to a talk given by Meeks last Friday, where he described much of his work on constant mean curvature and minimal curvature surfaces, and I realized Wikipedia doesn't even have any articles about Delaunay surfaces or constant mean curvature surfaces, which seems quite odd. This seems like a good starting point.

Thanks. --XediTalk 03:47, 27 January 2009 (UTC)


 * Yes they do need a lot of work. EG-Models and The Scientific Graphics Project might be a good resources for CMC surfaces. --Salix (talk): 08:36, 27 January 2009 (UTC)

College Math Help!
A balloon of total mass 680kg is descending with a constant accelearaiton of 0.4 m/s2. Find the upthrust on the ballon. When the ballon is moving at 1.5m/s, enough ballast is released for the balloon to fall with a deceleration of 0.2 m/s2. Calculate

(a) how much ballast was releasd

(b) the time for which balloon continues to fall

MY PROBLEM My problem lies in the bold part of the question. When ballast is released, I assume that the mass of the total object in consideration has also changed. But solving with this in mind, it doesn't work. Answers to this question: 6528N   (a) 40kg    (b) 7.5s People, help a poor guy please! —Preceding unsigned comment added by 202.72.235.201 (talk) 14:20, 27 January 2009 (UTC)


 * You have to assume that the buoyancy of the ballon does not change when it releases ballast, even though its mass changes. Work out the buoyancy force B from the first part of the question, then work out the new mass m using B - mg = ma, then subtract new mass from original mass to find ballast. I think you also have to take g to be 10 ms-2 to produce the given answers. Gandalf61 (talk) 14:55, 27 January 2009 (UTC)

Thanks Gandal for your thoughts but I did the math exactly the way you thought, it did not work out. g = 10 m/s2 taken also. —Preceding unsigned comment added by 202.72.235.201 (talk) 15:44, 27 January 2009 (UTC)
 * Gandalf's method gives exactly the answers provided. Can you explain your working, so we can see what's going wrong? Algebraist 15:48, 27 January 2009 (UTC)

Bouyancy or uptrust equals 6528N as stated in (a) Let the new mass of this balloon be 'm' Weight = mg = 10m Upthrust = 6528-10m N This upthrust causes upward acceleration of 0.2 m/s2 thus: upthrust = m x (acceleration) 6528-10m = m x 0.2 m = 640 ? Woh: Ballast released = 40 kg! Algebraist, I can see my mistake but I did the math for like an hour, I can't see why that time I didn't get my answers! Thanks for your help though! —Preceding unsigned comment added by 202.72.235.201 (talk) 17:04, 27 January 2009 (UTC)

Don't laugh People for my stupid math problem, it was a silly mistake back then, I didn't read the question well, I took a = 0.5 for 0.2. Got to sleep more and clear up my brains!

Math Problem Resolved! How do we get this section removed now! —Preceding unsigned comment added by 202.72.235.200 (talk) 17:16, 27 January 2009 (UTC)
 * Reading the question carefully is always a good first step - we've all been there! --Tango (talk) 17:24, 27 January 2009 (UTC)

why not fermats last conjecture
if he didnt prove it, why isnt it fermats last conjecture —Preceding unsigned comment added by 82.120.111.130 (talk) 16:11, 27 January 2009 (UTC)
 * He claimed he had a proof (that didn't fit in the margin) and people gave him the benefit of the doubt. Chances are, he was mistaken (it seems unlikely that nobody would be able to reproduce the proof after hundreds of years if it had existed - the only proof we have involves mathematics Fermat couldn't possibly have known about). --Tango (talk) 16:15, 27 January 2009 (UTC)
 * After all in maths it is quite common that the "theorem of X" has not been proved by X.--pma (talk) 16:58, 27 January 2009 (UTC)
 * Seems to be the case more often than not, as far as I can tell. --Tango (talk) 17:23, 27 January 2009 (UTC)
 * At this point in the thread it's obligatory that someone mention Stigler's law of eponymy (due to Merton). Algebraist 17:25, 27 January 2009 (UTC)
 * The proof uses algebraic geometry. -- PS T  12:01, 28 January 2009 (UTC)
 * Yes, it does, did you have a point? --Tango (talk) 22:31, 28 January 2009 (UTC)

Wake Angle
This page states that the 'angle of the wake of a body moving steadily in deep water is always 2arcsin(1/3)'. How is this result derived? Angus Lepper(T, C) 16:18, 27 January 2009 (UTC)


 * The angle is called the "Kelvin wake angle" - there is a derivation here. Gandalf61 (talk) 10:16, 28 January 2009 (UTC)


 * Thanks. Angus Lepper(T, C) 21:23, 28 January 2009 (UTC)

Co-ordinate geometry
Please consider the following....

Supposing y=X^2+3X+1 is an equation...we can draw y against x. Now, if, we draw two tangents at the points where the curve cuts the x-axis and also draw two normals through the same points. Now, lets assume that the two normals meet at A and the two tangents meet at B. My question is.....will the point A be a reflection of point B at x-axis and vise versa or the impression is wrong. I ask this qustion since I rhink that the tangent is a reflection of the normal and vise versa. Please, help me out. —Preceding unsigned comment added by 123.49.43.236 (talk) 17:33, 27 January 2009 (UTC)
 * For a quadratic, that would only be the case if the slopes of the tangents were 1 and &minus;1 respectively, which they aren't for your function. -- Jao (talk) 17:43, 27 January 2009 (UTC)
 * You can check this by noting that reflection in the x-axis will negate the slope of a line, while the slope of the normal line is the negative of the reciprocal of the slope. Thus to have the normal equal to the reflection of the tangent, you must have that the slope of the tangent is its own reciprocal. As Jao says, this implies the slope of the tangent must be 1 or &minus;1.


 * To find all the parabolas for which this occurs, we can translate our parabola to be symmetric about the y-axis, so that its equation is of the form y=Ax2-C. Then the two roots are at ±r, where $$r=\sqrt{C/A}$$ (which you can check with the quadratic formula). Then solving the equation y′=2Ax1= at r reduces to 4AC=1 (which happens to be the discriminant of our quadratic polynomial). Thus the general solution for symmetric parabolas is y=Ax2-(1/4A). Tesseran (talk) 17:59, 27 January 2009 (UTC)

saving time
Joe wonders if he can reduce the times it takes for him to test whether a coin is fair by stopping after 500 of his 1000 flips and seeing if a certain ratio isn't met, or doing the same in n segments. —Preceding unsigned comment added by 82.120.227.157 (talk) 19:00, 27 January 2009


 * It sounds like you want Sequential probability ratio test (SPRT). Sorry didn't read it all as it seemed to ramble and had lots of code in it. there's an acronym for that too tl;dr Dmcq (talk) 09:51, 28 January 2009 (UTC)


 * I reduced the problem.

I didn't follow Joe's programming either, but consider this. Knowing the probability of tossing head, $p$, say    $p$=1/2, Joe can forecast the number of heads,  $i$, after tossing $n$ times. The distribution of $i$ is the binomial distribution. The mean value of $i/n$ is $p$, and the standard deviation is $\sqrt{p(1&minus;p)/n}$. But this is not Joe's situation. Joe's situation is the reverse one: observing the number of tosses, $n$, and the number of heads,  $i$, and wanting to estimate the probability  $p$. The distribution of $p$ is the beta distribution. The mean value of $p$ is not $i/n$, but rather $(i+1)/(n+2)$, and the standard deviation is $\sqrt{P(1&minus;P)/(n+3)}$, where $P$ = $(i+1)/(n+2)$. If the mean value is more than 3 standard deviations away from 1/2, Joe may conclude that the coin is significantly unfair. Joe can never make sure it is completely fair. Bo Jacoby (talk) 12:58, 28 January 2009 (UTC).

hey Joe, I liked the story however ;) pma (talk) 14:02, 28 January 2009 (UTC)


 * Yes you need to decide what you mean by fair and how sure you want to be that it fits your criteria before you rush off testing it. Sequential analysis is all about cutting the cost of the testing once you've decided on your criteria. It's like those medical trials where they abandon them early if they find too many bad side effects. They could also stop if the effect is obviously good but they normally like to just continue on with the original scheme for that and get better data. Dmcq (talk) 14:16, 28 January 2009 (UTC)
 * By the way I first came across this sort of thing in a book 'Facts from Figures' by M. J. Moroney which I read whilst I was still at school. I though it was very good, I don't know if there is anything like it around now. Dmcq (talk) 14:26, 28 January 2009 (UTC)

you're right (dmcq) I didn't define what isn't fair. if not fair means it favors heads by at least 3%, can I save any flips but retain my confidence level by not necessarily doing 1000 flips if certain criteria are met? what is the optimal places and criteria to set my early-stops at? —Preceding unsigned comment added by 82.120.227.157 (talk) 14:57, 28 January 2009 (UTC)
 * I did define "fair" though. "fair", the default option (one I want to VERY rigorously eliminate)  means that a 50/50 coin produces the observed experimental results (more than 1 in 100,000 times).  So if an experimental result is something that a fair coin (as simulated) produces 3,953 times in one iteration of 100,000 runs, for that test the coin would be considered "fair", since I want to produce experimental results that a fair coin produces less than 1 in 100,000 times -- ie I can bet my life that the coin observed making those experimental results is not behaving as a fair coin does -- even 1 in 100,000 times.  —Preceding unsigned comment added by 82.120.227.157 (talk) 15:02, 28 January 2009 (UTC)

If Joe want 1 in 100000 security that the coin is unfair, he must test that the mean value of $p$ is more that 4.4172 standard deviations away from 0.500. (See normal distribution). Bo Jacoby (talk) 16:31, 28 January 2009 (UTC).


 * can splitting the 1000 flips into two groups and stopping if the first one fails instead of waiting for the total count to reach 1000 help reduce the average number of flips it will take to fail (as it probably will, since he is assuming the coin is probably fair). what abaout n groups?  does it help?  —Preceding unsigned comment added by 82.120.227.157 (talk) 16:57, 28 January 2009 (UTC)


 * I guess what I'm really asking is what the least many AVERAGE flips is that it takes to "prove that the mean value of p is more than 4.4172 standard eviations away from 0.500". Is the average better than the wrost-case ?  (where you have to do the whole experiment) or is there no way to streamline it that way?  —Preceding unsigned comment added by 82.120.227.157 (talk) 17:03, 28 January 2009 (UTC)

Joe may never prove that the coin is unfair. So the worst case is infinite. The average case depend on the unfairness of the coin. The algorithm is
 * $i$ := 1;
 * $n$ := 2;
 * repeat
 * make a group of flips;
 * $i$ := $i$ + number of heads in the group;
 * $n$ := $n$ + number of flips in the group;
 * until (P&minus;0.5)2 > 4.41722 $P$.
 * until (P&minus;0.5)2 > 4.41722 $i/n$.

Bo Jacoby (talk) 00:31, 29 January 2009 (UTC).

Evaluate trigonometric definite integral
$$\int_{0}^{\pi/4}\sin^4(x)\cos^2(x)dx$$

I tried using power reduction identities, but that just left a big mess of sin's and cos's. Pythagorean identities didn't really work either. Just need a hint on what identities to use, no need to do the entire problem.--Yanwen (talk) 20:24, 27 January 2009 (UTC)


 * $$\begin{align}

& \int_0^{\pi/4}\sin^4(x)\cos^2(x)\,dx = \int_0^{\pi/4} (\sin^2 x)^2 \cos^2 x\,dx = \int_0^{\pi/4} (1-\cos^2 x)^2 \cos^2 x \,dx \\ & = \int_0^{\pi/4} (1 - 2\cos^2 x + \cos^4 x)\cos^2 x \,dx = \int_0^{\pi/4} \cos^2 x - 2\cos^4 x + \cos^6 x \,dx \end{align}$$ etc. Do the power-reduction identities you've looked at fail to work with this? Michael Hardy (talk) 21:11, 27 January 2009 (UTC)
 * Otherwise, you can integrate by parts repeatedly, starting from $$\textstyle\sin^4(x)\cos(x)=\left(\frac{1}{5}\sin^5(x)\right)^'$$. Is it clear how to go on?--pma (talk) 21:40, 27 January 2009 (UTC)


 * For an unpleasant but totally mindless approach, you can always just express sin and cos in terms of ex, multiply out, and integrate. Algebraist 21:53, 27 January 2009 (UTC)
 * It's our great utopia, not to have to think! ;) Also, in all cases, starting with a change of variable: $$2x=y$$, will lower the degree and make it easy. --pma (talk) 23:10, 27 January 2009 (UTC)
 * Integration by parts works great. Substitution is also good. Thanks!--Yanwen (talk) 03:31, 28 January 2009 (UTC)
 * Welcome! On these lines, you may like the computation of $$\scriptstyle I(n):=\int_0^{\pi}\sin^n(x)\,dx$$, an integral arising in the computation of the volume of the euclidean n-ball. Integrating by parts you find $$I(n)$$ as a certain rational multiple of $$I(n-2)$$. Since $$I(1)=2$$ and $$I(0)=\pi$$, you get a product formula for $$I(n)$$. There is another interesting fact. You can observe that: 1) $$I(n)$$ is a decreasing sequence; 2) $$I(n+2)/I(n)$$ converges to 1, therefore so does $$I(n+1)/I(n)$$; 3) $$I(2n+1)$$ is a rational number; 4) $$I(2n)$$ is a rational multiple of $$\pi$$. As a consequence, $$\scriptstyle \pi I(2n+1)/I(2n)$$ is a rational approximation of $$\scriptstyle\pi$$. It's the Wallis product. --pma (talk) 13:55, 28 January 2009 (UTC)