The problem goes like this :       There is a circular field of radius = $r$ m (say 7 m), and a cow is tied with a rope of $x$ m on the circle. If cow grazes all the grass which is in its range then the cow would have grazed half of the total grass on the field. (Grass is uniform all over the field). Find the length of the rope (i.e., $x$).   I tried various methods like completing the circle from the arc made by cow, taking half the area of the field and finding the radius of another circle etc. but was unable to find the answer. Actually, I've no idea right now how to proceed.    [Note] This is not my HW but a question which was given by a teacher to my class.
What length of rope should be used to tie a cow to an exterior fence post of a circular field so that the cow can only graze half of the grass within that field?  updated: To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field.
Is $7$ the only prime followed by a cube? I have removed Windows 7 using GParted, and want my computer to boot straight into Ubuntu, but I still get the dual boot screen, how can I fix this? My months and days in my File Manager are in German, how do I change that? On the photo you see 'Dez' : It means Dezember (German for December). I also saw one day Mittwoch (German for Wednesday). How do I change that German issue? This is only my File Manager and not completly but partially. Weird. Rerouting stdin and stdout from C How can I change the default program installation directory in Windows 7? Windows 7 is installed on my C drive, which is quite small.  I am very tired of instructing new programs to put their files on my larger D drive during installation; I would like to change the default drive.   says that you can use a registry hack, but I am giving Microsoft the benefit of the doubt and naively assuming that a configuration option exists somewhere.  It's 2010... do I really have to hack my registry to make a simple tweak like this?  Also, there's a  that explains how to move the "Users" directory and create a symlink, which could also work.  However, at the moment I have some apps in C:\Program Files, some apps in C:\Program Files (x86), and some apps in the corresponding folders on D:\, so it would be a hassle.  Also, my small OS boot drive is a 10k RPM WD Raptor, and I feel like that probably gives a speed boost to apps installed on it that need to read &amp; write to their directories a bunch.  I wonder if it actually matters. Functions of Independent Random Variables Is the claim that functions of independent random variables are themselves independent, true?   I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of a normal distribution, but I have not been able to find justification for it. It seems that some authors take it as given but I am not certain that this is always the case. Very basic question: how do I write a short literal in C++?  I know the following:   2 is an int 2U is an unsigned int 2L is a long 2LL is a long long 2.0f is a float 2.0 is a double '\2' is a char.   But how would I write a short literal?  I tried 2S but that gives a compiler warning. I would like to count the number of unique selections from a column that contains multiple selections from a drop down list.  For example, column B3 contains,     Monday, Tuesday, Wednesday   The count function returns a value of 1 instead of 3 - is there any way to count the three days distinctly? Thank you!  Multiple selections from a drop down list was made possible using the VBA code from: Javascript infamous Loop issue? Including Bitmap (.bmp) I can't include bitmaps with pdflatex. Here is my mwe:  \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel} \addto\extrasturkish{\uccode`i=\string"9D \lccode`I=\string"19 } \addto\noextrasturkish{\uccode`i=`I \lccode`I=`i }   \usepackage{graphicx}  \begin{document}  \shorthandoff{=}  \includegraphics[width=5cm,height=5cm]{000.bmp} \end{document}   Gives the error Unknown graphics extension: .bmp. Yesterday I was filming a scene with my phone but suddenly the application crashed and when I try to open the m4v generated it says that is corrupted.  I have downloaded the video files (around 36MB) on my PC and I am trying to open and repair it. Can you suggest me a way to do it?  FFMPEG info  As suggested in comments I try check what FFMPEG says me about that file with this command  ffmpeg.exe -i VID_20120425_144739.m4v -c:v copy -c:a copy out.mp4 2&gt; log.txt   Here it is the result:  ffmpeg version N-40126-ga4b58fd Copyright (c) 2000-2012 the FFmpeg developers   built on Apr 26 2012 03:29:50 with gcc 4.6.3   configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-runtime-cpudetect --enable-avisynth --enable-bzlib --enable-frei0r --enable-libass --enable-libcelt --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libfreetype --enable-libgsm --enable-libmp3lame --enable-libnut --enable-libopenjpeg --enable-librtmp --enable-libschroedinger --enable-libspeex --enable-libtheora --enable-libutvideo --enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx --enable-libx264 --enable-libxavs --enable-libxvid --enable-zlib   libavutil      51. 47.100 / 51. 47.100   libavcodec     54. 15.100 / 54. 15.100   libavformat    54.  3.100 / 54.  3.100   libavdevice    53.  4.100 / 53.  4.100   libavfilter     2. 72.100 /  2. 72.100   libswscale      2.  1.100 /  2.  1.100   libswresample   0. 11.100 /  0. 11.100   libpostproc    52.  0.100 / 52.  0.100 [mov,mp4,m4a,3gp,3g2,mj2 @ 003fc940] moov atom not found VID_20120425_144739.m4v: Invalid data found when processing input   Latest two lines are displayed in red. This question is the flip-side of the commonly asked food temperature safety concerns. I'd like to know, if a meat-based soup stock (chicken/pork/beef/etc) were to be held constantly at a safe hot temperature of 165F/73C-180F/82C, how long could it stay at this temperature and still be considered safe to eat?  Obviously, it would reduce over time, but suppose it's reconstituted with water regularly so that it could continue being held... Is there ANY sort of time limit (even completely unreasonable spans like, months, years, etc)? Why does Beamer not change the itemize symbol when using \setbeamertemplate? This code:  \documentclass[10pt]{beamer} \setbeamertemplate{itemize item}[circle] \setbeamertemplate{itemize subitem}[triangle] \setbeamertemplate{itemize subsubitem}[square]   \begin{document} \begin{frame} \begin{itemize} \item first level     \begin{itemize}     \item second level         \begin{itemize}         \item third level         \end{itemize}     \end{itemize} \end{itemize} \end{frame} \end{document}    produces this output:    I expected the symbol of the first level item to be a circle, not a square. Is there an error in my code? The log file was too long to include in the question, so I . I am running Ubuntu Gnome 16.10 (64bit). Gnome-shell version 3.20.4.   When I start my computer, I eventually see GDM3 offering a login prompt, as expected.   After I login, my monitors flash a little (like they're unplugged and then replugged in), and then I am brought to the gnome 3 desktop.   However, if I hit  CTRL + ALT + F1, I am brought back to GDM3, offering a login prompt.   If I then hit CTRL + ALT + F2,  I am brought back to my Gnome 3 desktop.  So it seems like GDM3 continues to run in TTY1 while the desktop runs in TTY2.   Questions   Is this how things are supposed to work? I thought the desktop was supposed to use the same TTY (or whatever you call it) as the display manager.  Am I using more graphical resources by having things setup this way? Is there a way to have GDM3 pass off the display to Gnome Desktop after I login? Is there any advantage do doing #3? Is it possible to find all php files within a certain directory that have been modified on a certain date  I'm using   find /var/www/html/dir/ -mtime -28 | grep '\.php'    to get files modified within the last 28 days, but I only need files that have been modified on the following date: 2011-02-08 What "exactly" happens when someone turns to the Dark Side? When I saw the original Star Wars saga as a kid, the concept of the Dark Side imprinted in me was quite simple. You did a bad thing, so you're a bad person so you go to the Dark Side and keep doing bad things.  Now, obviously this doesn't hold true in reality. In my mind it is perfectly reasonable for Luke to kill the Emperor and Vader and then just flee the Death Star and rejoin the Alliance. After all they are at war, those are enemy officers that have captured him and are taunting him by threatening with death of all loved ones and the whole operation was intended to kill the Emperor anyway.   The same would hold true for Anakin. He could have killed Mace Windu to save the Chancellor and Padme but that doesn't mean he's going to then go mad and kill all Jedis including children. It doesn't add up from a merely psychological point of view. He had a set of objectives and that made him want to keep the Chancellor alive (quite valid by the way, Mace Windu is quite clumsy on that) but just after killing him he goes into "Yes master I'll do whatever you say" which from a normal point of view will NEVER ever make sense.  In both cases it seems like the assumption is once you use the Dark Side you completely turn to the Dark Side and become a different person that you were before. That would be coherent with Anakin "dying" and Vader being born.  Is that the way the Dark Side works? Is there any canon information about what happens when someone turns to the Dark Side? Specifically to whether they become a different person or how he or she actually changes?  Note: Vader is angry after he finds out Padme is dead, even though he's already turned.  Example of what I mean. In Buffy they explain how people that turn into Vampires lose their soul and it's taken over by a demon, and thus they are, in all effects, a different person with their memories. save as a video as .mp4 So I made an intro using animation and it saves it as a .blend and I want it so save as a video like .mp4 or something like that. I've tried changing the codec because I tried opening the intro with windows media instead of blend. I didn't try all the codec's but I tried some of them. Diamond moderators  . In the second section from the top—entitled &quot;Ask questions, get answers, no distractions&quot;—that takes us to an opportunity to select one of a page full of questions from the site as the example question used on the about page. The selections currently available on physics are a little idiosyncratic. Is there some rhyme or reason to these? I suppose that we want a good example question to:  Be accessible to the intended audience (perhaps even to the newest and most naive person in that audience) Be judged at least decent by the community (positive score) Have more than one answer which are also judged acceptable (again positive score)  What else? How does a question end up on the list? Is it possible to have connecting loop lines (like algorithm2e) in algorithmic? I am using the algorithmic package to typeset an algorithm. I would like to have connecting lines between the beginning and end of a loop or conditional like in the algorithm2e package. Is there an easy way to accomplish this, or am I better off switching to algorithm2e? is that function must be constant under the following conditions I'm talking about complex function   $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition $|f(\frac{1}{n})| \leq \dfrac{1}{2^n}$   is that function must be constant ?