You guys are right, sorry. In my head I was assuming the full radius of the rope to be available minus the area of the silo, but as stated, that's not correct. Apologies.
[QUOTE=Falubii;42906133]This problem could be greatly simplified if this oppressive farmer allowed his cows to be free range.
Edit:
Also, carets are lame. What are the chances this forum will ever get superscripts/subscripts?[/QUOTE]
There used to be sub/super scripts but Garry decided not enough people used them (for sensible purposes) and so they were removed.
There is a [url=https://chrome.google.com/webstore/detail/tex-the-world-for-chromiu/mbfninnbhfepghkkcgdnmfmhhbjmhggn]plugin[/url] for Chrome (and probably for FF as well) that allows you to easily embed [;\LaTeX;]. I tend not to use it though, since I have no idea how many people would actually have it. [sp]Also I'm too lazy to mark it up in proper TeX[/sp]
I love the LaTeX chrome plugin but yeah, I assume not enough people here use it for it to be worth typing in. :v:
That's why I love sites like math.se with LaTeX rendering built in.
[QUOTE=Joey90;42905394]First split the problem in half across the middle, then break it up again:
[img]https://dl.dropboxusercontent.com/u/4081470/silol.png[/img]
The green section is easy - quarter of a circle radius [;\pi r;] has area
[;\frac{1}{4} r^2 \pi^3;]
The blue section we need to do an integral. After careful consideration, the best thing to integrate over is actually the piece length of rope [i]not[/i] pressed against the silo, marked [;x;] below:
[img]https://dl.dropboxusercontent.com/u/4081470/silol2.png[/img]
Set alpha to be the angle round from the centre. Now consider 'unwinding' by [;dx;], this corresponds to a change in angle of d alpha. Conveniently, since [;x;] and [;\alpha;] are directly proportional, [;dx = r d \alpha;]
We want to calculate the yellow shaded area, and we do this by approximating it as a sector of a circle. In which case it has area
[;\approx \frac{1}{2}x^2 d \alpha = \frac{1}{2r}x^2 dx;]
Now we can integrate over [;x;] from [;0;] to [;\pi r;]:
[;\frac{1}{2 r} \int_0^{\pi r} x^2 dx;]
[;=\frac{1}{6 r}\left[ x^3 \right]_0^{\pi r};]
[;=\frac{1}{6}r^2 \pi^3;]
adding the two together gets you
[;\frac{5}{12}r^2 \pi^3;]
then double it (since we've only done half) to get
[;\frac{5}{6}r^2 \pi^3;][/QUOTE]
I really appreciate the help you guys offered, unfortunately my power went out just after I posted this and wasn't able to see it until the assignment was already due. Power was out for a few days. Luckily I managed to catch the professor before class and he somewhat explained it. Same way as this user did. I'm not quite sure why he assigns us these problems, knows that pretty much the entire class has to see him at office hours, then doesn't even put it on the test, but yet grades us on the homework assignment. Bleh
[editline]22nd November 2013[/editline]
The answer my professor gave me was difficult to understand. I understand your way much better. He was including Sin and Cos which confuses me, but even on the material I already understand, he seems to not understand a good way to convey his knowledge. He was confusing me on material I already knew, when I would do it in like 4 lines, and he would do it in double that, still come to the same answer...
Got this problem in a practice coursework:
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.04.21.png[/img]
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.05.41.png[/img]
Is my proof valid? (Forgive the formatting, I'm new to LaTeX)
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.06.52.png[/img]
[QUOTE=sambooo;42962174]Got this problem in a practice coursework:
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.04.21.png[/img]
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.05.41.png[/img]
Is my proof valid? (Forgive the formatting, I'm new to LaTeX)
[img]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202013-11-24%2001.06.52.png[/img][/QUOTE]
Afraid not - the step
[;a^2 \nmid b^2 \implies a^2 \mid n;]
isn't in general valid. I think you're saying that [;a^2;] either has to divide [;n;] [i]or[/i] [;b^2;]. This would only work if [;a^2;] was prime... (Which is certainly isn't).
What you should do instead, is try and prove what it tells you to!
I think the clearest way to do this is as follows, for a prime [;p;], write [;a;] and [;b;] as:
[;a=p^r c,\; b=p^s d;] where [;c;] and [;d;] are coprime to [;p;]
Now
[;b^2 \mid a^2 \implies p^{2s} d^2 \mid p^{2r} c^2 \implies p^{2s} \mid p^{2r} c^2;]
But now since [;c;] is coprime to [;p;], we get
[;p^{2s} \mid p^{2r};]
From which we can deduce
[;2s \leq 2r \implies s \leq r;]
i.e. the power of [;p;] dividing [;b;] is less than or equal to the power of [;p;] dividing [;a;]. Since this is true for all primes, and we have uniqueness of prime decomposition, this means [;b;] divides [;a;].
Now we have that result, proving what we want is immediate. If [;a^2 = n b^2;] we know
[;b^2 \mid a^2;]
so from above we get
[;b \mid a \implies \frac{a}{b} \in \mathbb{N};]
as required.
Is there a plugin for firefox that makes that post make sense? I can't read latex
[editline]25th November 2013[/editline]
Also can I not just add in a step that says any factor p^n (p prime, n natural) of a is not a factor of b and thus any factor p^2n (which is all of them since all prime powers would be doubled) of a^2 is not a factor of b^2 and thus must be a factor of n?
[QUOTE=sambooo;42972620]Is there a plugin for firefox that makes that post make sense? I can't read latex
[editline]25th November 2013[/editline]
Also can I not just add in a step that says any factor p^n (p prime, n natural) of a is not a factor of b and thus any factor p^2n (which is all of them since all prime powers would be doubled) of a^2 is not a factor of b^2 and thus must be a factor of n?[/QUOTE]
Possibly this [url]http://thewe.net/tex/[/url] will do it? I don't know how significant the performance hit is though (maybe turn it on only if you know it will do something)
I think you probably can do it that way. The hard bit about the problem isn't really seeing [i]why[/i] it's true (it's obvious right?) but actually proving it rigorously. It's essentially the same method, just you are assuming that a and b are coprime and I wasn't (but neither method is clearly better than the other).
As an aside, what do you guys use for editing latex? I'm using LyX since it seems to be popular but not being immediately able to edit the actual latex code is really annoying me. I can see it the whole time but can't type in its box.
Also, do any of you have experience with Maple and figuring out its terribly unhelpful error messages? I'm a programmer so it should come easily but I'm not the software whisperer so it doesn't
TeXMaker, works like a charm compared to TeXnicCenter( though I'd love to be able to detach my render screen from my text editor).
If you chose Maple yourself, change it to Mathematica or Matlab depending on what you want to do. I find it a horrible piece of shit that has no use in academics.
My course a couple of years ago gave a pdf with common error messages and the probable mistake, but I don't think I still have access to those documents (and it's in Dutch)...
[URL="http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=ErrorMessageGuideOverview"]
Anyway, here's my (probably futile) attempt at helping you.[/URL]
I use LyX for simple stuff like homework and TeXworks for big stuff since it's more flexible.
I've only ever really tried TeXmaker and it seemed to do everything that I wanted so I never got round to experimenting with others.
I didn't choose Maple, a piece of coursework for one of my modules has to be done in it. I (wrongly) assumed that f(x)(x-pi/3) would evaluate to multiplying those two functions together because surely it'd error otherwise, right? That line went through fine, no warning or anything. Then the call to plot(f(x)(x-pi/3)) comes along and the shitty tool tells me that the function isn't defined over the range I specified. The real issue was a missing asterisk, nothing to do with what the message told me at all. Fuck Maple.
As for LaTeX stuff, I've got LyX and TeXmaker installed and I'll figure out which I like more later since I'm not likely to actually need either soon. Thanks guys.
This could probably go in the science thread, but this one seems more active so...
I was looking at a physics quiz my little brother took and I couldn't figure out a problem.
[quote]An amusement park ride called the "Cliffhanger" consists of a cylindrical wall that spins its riders in a circle and forces them against the wall of the ride. When the ride is up to operating speed the floor drops out from beneath them and they are supposed to stick to the wall. What is the minimum tangential velocity the ride must spin if it is assumed that the largest person that will take the ride has a mass of 140 kg and the radius of the wall is 3.0 meters? Assume that the coefficient of friction between the rider and the wall is zero.[/quote]
Now I might be missing something here, but how is the person supposed to be pinned to the wall when the coefficient of friction is zero? It's definitely possible that it's an error (it is a high school quiz).
[QUOTE=Falubii;42991617]This could probably go in the science thread, but this one seems more active so...
I was looking at a physics quiz my little brother took and I couldn't figure out a problem.
Now I might be missing something here, but how is the person supposed to be pinned to the wall when the coefficient of friction is zero? It's definitely possible that it's an error (it is a high school quiz).[/QUOTE]
I agree with you. If the coefficent of friction is 0, then the force of friction is 0. Personally, I would point it out, and then do the problem assuming mu is 1, as I think that would be a pretty good guess as to what the teacher was thinking.
Or ask for a conical wall :v:
It was a multiple choice quiz, and it seems one of the answers corresponds to a μ of 1.
Edit:
Did I miss something? Should I be doing stand-up instead of physics?
Select quotes from various lecturers at my Uni-
[b]Professor Y, Analytical Mechanics-[/b]
"I've never read Newton's proof because it's beyond human understanding"
"Like most french people, d'Alembert also had a weird, funny last name"
"And here we have an expression for potential energy, kinetic energy and some more shit like that"
"Already, Newton knew this was a shit proof"
"These are all very good questions, so I will ignore them"
"We did this the last lecture, but I like it so much so lets do it again"
"Now we have to place that expression into the equation, but i'm not that brave so i'll just write down the final result"
"Making these kinds of calculations in public is indecent"
"What does this teach us? That we're all old"
[b]Professor A, Waves-[/b]
"Now I've convinced you that this equation is true. You! Yes, You! Are you convinced?" "Not really..." "Forget it, I didn't ask you, guy behind him, are you convinced?"
Immediately after a break - "You came all the way to the bathroom to ask me about the test?"
"Why are you checking your phone? Can anything be more interesting than me?"
"Infinity is a place far, far away"
While looking at the stuff he just wrote down on the board- "What an explanation!" (kisses his hand and touches his forehead)
[b]Professor E, Complex Functions-[/b]
"The negative real axis, is the axis of evil, ahmadinejad lives there!"
E: "Now tell me, does a function like this exist?"
Student: "Yeah, you just integrate both sides."
E: "I like the first part, the second not so much."
Student: "But you really can integrate both sides and it'll work itself out, what's the problem?"
E: "Hey, if you wanna fight about it, take your shirt off and we'll go outside!"
[b]Professor Y, Probability-[/b]
"If you don't know what Laplace and Fourier transforms are, they were both french"
[b]Professor M, Differential Equations-[/b]
"Now lets ask a question- Why is it called "Dirichlet kernel"? Because!" (continues teaching)
"We will look at the precious stones of infinitesimal calculus, there's plenty of stones, a whole bunch of them!"
Today I've discovered that finding the determinant of a variable based 4x4 matrix is an exercise in tedium.
Half an hour to discover that, in fact, x(y(sz)) gives me absolutely no valuable information whatsoever.
[QUOTE=Mr. Bleak;43030996]Today I've discovered that finding the determinant of a variable based 4x4 matrix is an exercise in tedium.
Half an hour to discover that, in fact, x(y(sz)) gives me absolutely no valuable information whatsoever.[/QUOTE]
A lot of the shit you do as a math major pre-your-first-proofs-class is an exercise in tedium.
[url]http://arxiv.org/ftp/math/papers/0309/0309103.pdf[/url] My favourite proof of all time
[QUOTE=Glorbo;43029814]Select quotes from various lecturers at my Uni-
[/QUOTE]
[b]Professor S, Analytical Geometry, Linear Algebra, Calculus 2 & 3 (oh, shit, I had him for too many classes)-[/b]
"You take the determinate of blah; and multiply by the function of 'who cares' and get some bullshit."
Also, I like your Prof. E just for that Ahmadinejad crack.
[B]Professor B, Combinatorics and Cryptography[/B]
"Well shit in my mouth."
[QUOTE=JohnnyMo1;43061903][B]Professor B, Combinatorics and Cryptography[/B]
"Well shit in my mouth."[/QUOTE]
That probably won't sound much less weird even in context.
I don't even know the context. I was in office hours with my topology professor, next door. :v:
So I took a math test today that had a problem that really stumped me (because I didn't remeber my trig identities)
It was in a high school calculus course (IB).
The problem went like this:
Suppose sin(100) = m. Find, in terms of m, these expressions:
cos(100)
tan(100)
sin(200)
Now, the second two were very easy to do but they depended on the first one, which I completely forgot how to do.
One of my friends had the idea to put down cos(100) as the derivative of m (which I guess it technically is) and then solve the rest based on that.
I'm not sure if that's valid, but if isn't, what is?
Use the fact that cos^2(x)+sin^2(x)=1? Unless I'm missing some other requirement...
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