Down to zero, if the smallest size of a simple container is less than the diameter of the sphere for example, even if you had cubic miles of space in a tube the diameter of five inches, you can't fit a single six-inch sphere in it.Īlthough each void is relatively small compared to the size of a sphere, there are lots of them – remember, each sphere touches 12 other spheres! –, totaling about 26% of the container volume. When the container is not immense compared to the spheres, more volume (relatively speaking) is wasted between the spheres and the container walls, so the real $N$ you can achieve is less than what the above predicts. The larger the container is compared to the size of the spheres, the closer you can get to the limit given above. The rest of the volume is "wasted" in the small voids between the spheres, and in the voids between the spheres and the container walls. Finding the formula itself is easy enough to do, the real problem is finding out where the numbers come from ( $N \, \le \, \frac$$spheres in the container. Would anyone mind clarifying why the maximum density is 74%?Įdit: I had some great responses, thank you very much for them! A few problems though: My own knowledge is only equivalent to an advanced class high schooler, so that's probably why this goes over my head. However, I have yet to really understand how they derive this number. the maximum density the sphere's can occupy in the close packing method is around 74%. From what I've gathered from previous Stack Exchange questions etc. However, having my own doubts about the correctness of the method I did some research on my own and came across close packing. And so this is sometimes the event in question, right over here, is picking the yellow marble. The probability of picking a yellow marble. So they say the probability- I'll just say p for probability. Initially my friend was working with a crate method, namely turning the spheres into cubes whose length was equal the sphere's diameter, and finding out how many cubes fit (with some estimates, this is easy enough). Find the probability of pulling a yellow marble from a bag with 3 yellow, 2 red, 2 green, and 1 blue- I'm assuming- marbles. Obviously, since a soccer ball is a sphere, the resulting question is the title. The assignment is: Find how many soccer balls fit inside a cylindrical building. A jar contains 5 blue cubes, 4 blue spheres, 2 green cubes, and 7 green spheres. You can often write recurrence relations for counting problems like this, but here it’s far more work then needed, and you seem to have found a simple solution.So this is on the behalf of another friend for a school assignment. If you reach the right-most face in the red area, you’ve run out of red candies first, so to answer your question, you want to know how many paths (going only right, up, or away) there are from the origin to the upper-right box that reach the red area.Īlternatively, you would want to know what fraction of the $\frac$ sequences do not correspond to paths that lie within the following figure. In the picture below, a candy-sequence would correspond to a path of adjacent cubes from bottom left to top right that moves right, up, or away from the viewer. Note, this only holds because the balls are being randomly selected. So the probability of getting a blue ball is 7/21 or 1/3. If you think of picking red, blue, and green as corresponding to taking steps in the $x$, $y$, and $z$ directions on a three-dimensional lattice, each way of drawing candies corresponds to a path from $(0,0,0)$ to $(10,20,30)$ of steps that go one unit in a positive direction each. There are 7 blue balls and 21 balls altogether. The following is almost certainly overkill, but what the heck. 4 Blue Cubes Second container: 4 Red Cubes 3 blue Cubes There are 4 outcomes and they are as followed: (R,R), (R,B), (B,R) And (B,B) The question is asking about the probability of one cube is red out of the 2 containers.
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Information on AppleInsider.Ĭurrent Version: macOS Catalina 10.15.7 Update 2022-005, July 20, 2022. Previous Version: macOS Big Sur 11.7.5, MaSecurity content of macOS Big Sur 11.7.5. Information on MacRumors and AppleInsider. Information on AppleInsider.Ĭurrent Version: macOS Big Sur 11.7.6, ApSecurity content of macOS Big Sur 11.7.6. Previous Version: macOS Monterey 12.6.4, MaSecurity content of macOS Monterey 12.6.4. Information on AppleInsider, 9TO5Mac, and MacRumors.Ĭurrent Version: macOS Monterey 12.6.5, ApSecurity content of macOS Monterey 12.6.5. Previous Version: macOS Ventura 13.3, MaSecurity content of macOS Ventura 13.3. Information on AppleInsider, 9TO5Mac, MacRumors, and MacRumors. Some Security Updates are folded into minor updates instead of being released as separate updates.Ĭurrent Version: macOS Ventura 13.3.1, ApSecurity content of macOS Ventura 13.3.1. When an update marked Supplemental Update or Security Update is released, it is generally safe to install these shortly after the update is released, although I generally follow the same schedule and install such updates on the Friday after release.I generally install such updates on the Friday after release to give myself the weekend to further test and ensure my Mac is working as should from the update before the following week. When a minor update to a major operating system version is released (for Catalina, that would be 10.15.1, 10.15.2, 10.15.3, etc.), it is generally recommended to wait a few days after the update has been released before installing it to ensure no major issues have been noted.When a major operating system upgrade is released (such as macOS Catalina 10.15 in October), it is generally recommended to wait and install the upgrade after its first minor update is released (in this case, 10.15.1).General recommendations for installing macOS upgrades: OS X Mavericks to the present version of macOS are named after California locations. Mac OS X Cheetah-OS X Mountain Lion named major operating system releases after large cats. Major operating system upgrades for all future macOS releases are delivered through an installer downloaded from the Mac App Store (prior to OS X Lion, operating system upgrades were delivered through a DVD). Instructions for installing macOS updates can be found here. macOS Mojave-Present has moved operating system update installs back through Software Update in System Preferences. OS X Mountain Lion-macOS High Sierra installs operating system updates through the Mac App Store. Mac OS X Cheetah-OS X Lion installs operating system updates through Software Update in System Preferences. MacOS (formerly Mac OS X and OS X) is the operating system that powers Mac computers. Occasionally, Apple will make major changes to how app data syncs across devices over iCloud, and waiting to upgrade devices after the major operating system upgrades have been released for each device type provides for the smoothest upgrade experience. Instructions on updating each device are also included.įor users with multiple Apple devices (such as a Mac, iPhone, and iPad, etc.), I recommend waiting to install annual major operating system upgrades after Apple has released the major operating system upgrades for each device type. This is a regularly updated list of Apple operating system updates (grouped by device), along with guidance in determining when it is recommended to install each update. AKB2000014 Ongoing List of Apple Operating System Updates Tom Hough tomhbeatle1 16.1K subscribers Subscribe 2.9K 720K views 11 years ago The song is called the endless dream by 'YES'. Kelly was notable as the only female trucker featured in the series. From seasons 3 and 5, and 7 and 11, Kelly and her fellow drivers were followed along Dalton Highway with oversized loads. Lisa lives by one rule: challenge herself to be the best she can be in this male-dominated field. Lisa Kelly, sexiest pics (IRT) 5 She is eye candy. 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Photos is extremely easy to use, but it may not be perfect, as some duplicate photos not showing up has been a common issue on Ventura, also some Mac users reported that they can’t find all duplicate photos. You can forget about finding files or images that are similar as well. Finder is good at displaying all files, but it cannot just automatically delete duplicates with great accuracy. Are Finder and Photos are good enough for finding duplicates?Įven though Finder and Photos can be a great option to start with – They’re not ideal or as powerful as some third-party picks we’ve tested.You will see any duplicates automatically appear here as above image, and you can delete or merge them. In the Photos app, simply click Duplicates in the sidebar. Yes, Apple has added a nifty feature to macOS Ventura in the Photos app that detects duplicate photos and videos for you. Use Photos App (for macOS Ventura users).Even though you will still have to do the work manually – it won’t be hard to find duplicate files as they’ll have the same name and will be listed one under another. When you see a list, the last step you have to do is go through the list and find duplicate files. Finder will do all the work from there, and it will come up with a list of all files on your Mac. In there, you can set up the kind of file list you would like to look out for and the type of files you would like to list. It’s a mighty tool integrated into every Mac computer. All you have to do is open Finder and click File -> New Smart Folder. Many Mac users forget that Apple’s built-in apps have some particular options to help them find and list duplicate files in just minutes. Can my Mac find and delete duplicate files? Note: If you buy something from links on this page we may earn a commission at no extra cost to you.
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