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Motion problems: finding the maximum acceleration. Next lesson. Current timeTotal duration Optimization problems calculus. Video transcript Let's say that we have a sheet of cardboard that is 20 inches by 30 inches. Let me draw the cardboard as neatly as I can. So it might look something like that. So that is my sheet of cardboard. And just to make sure we know the dimensions, there's 20 inches by 30 inches.
And what we're going to do is cut out the corners of this cardboard.
And all the corners are going to be squares, and we're going to cut out an x by x corner from each of the corners of this piece of cardboard-- x by x. Over here, x by x, and then over here, x by x. And what we'll do is after we cut out those corners, we can essentially fold down the flaps. Let me draw the flap.
So you could imagine we can fold right there, we could fold right there, we could fold right there, and we would form a box.
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I guess you could imagine a box without a bottom to it, or you can view a box without a top to it. So if we were to fold everything up, we would get a container that looks something like this. Let me make my best attempt to draw it. So it would look like this. This is one flap folded up. You can imagine this flap right over here, if I were to fold it up like that, it now would look like this.
It would now look like this. The height of the flap is x. So this distance right over here is x. And then if I were to fold this flap, if I were-- let me do that a little bit neater. If I were to fold this flap right over here, if I were to fold that up, then it would look like this. It would look like that. And then I would fold that back flap up. So the back flap would look something like that. That would be the back flap. It would look something like that.
And then this flap over here would-- if I fold it up would look something like that. And then of course, my base of my whole thing, so this whole region right over here of my piece of cardboard, would be the floor of this box that I'm constructing. And what I want to do is I want to maximize the volume of this box.
I want to maximize how much it can hold. And I want to maximize it by picking my x appropriately. So let's think about what the volume of this box is as a function of x. Well, in order to do that, we have to figure out all the dimensions of this box as a function of x. We already know that this corner right over here, which is made up of when this side and this side connect when you fold these two flaps up, that's going to be the same height over there.
That's going to be the same height over there. The height of this box is going to be x. But what's the width? What is the width of this box going to be? Well, the width of the box is going to be this distance right over here. And this distance is going to be 20 inches minus not 1x, but minus 2 x's. So this is going to be 20 minus 2x. You see it right over here. This whole distance is You subtract this x, you subtract this x, and you get this distance right over here. So it's 20 minus 2x.
Now, the same logic, what is the depth of the box? What is that distance right over there? Well, that distance is this distance right over here. We know that this entire distance is 30 inches. If we subtract out this x and we subtract out this x, we get the distance that we care about. So this is going to be 30 minus 2x. So now we have all of the dimensions. So what would the volume be as a function of x? Well, the volume as a function of x is going to be equal to the height, which is x, times the width, which is 20 minus x-- sorry, 20 minus 2x times the depth, which is 30 minus 2x.
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Now, what are possible values of x that give us a valid volume? Well, x can't be less than 0. You can't make a negative cut here. Somehow we would have to add cardboard or something there. So we know that x is going to be greater than or equal to 0. So let me write this down. And what does it have to be less than?
Well, I can cut at most-- we can see here the length right over here, this pink color, this mauve color, is 20 minus 2x. This environment is practically indistinguishable from reality although scenes set within the Matrix are presented on-screen with a green tint to the footage, and a general bias towards the color green , and the majority of bluepills - humans connected to the Matrix - are unaware of its true nature. Most of the central characters in the series are able to gain superhuman abilities within the Matrix by taking advantage of their understanding of its true nature to manipulate its virtual physical laws.
The virtual world is first introduced in The Matrix. The Animatrix short film " The Second Renaissance " and the short comic Bits and Pieces of Information show how the initial conflict between humans and machines came about, and how and why the Matrix was first developed. Its history and purpose are further explained in The Matrix Reloaded. The series was filmed in Australia and began with 's The Matrix , which depicts the recruitment of hacker Neo into humanity's rebellion against sentient machines.
The film's mainstream success had backed up the initial idea of making a trilogy. They tell the story of the impending attack on the human enclave of Zion by a vast machine army. Neo also learns more about the history of the Matrix and his role as The One. The sequels also incorporate more ambitious action scenes and visual effects.
While making the Matrix films, the Wachowskis told their close collaborators that at that time they had no intention of making another one after The Matrix Revolutions. Jordan attached to star. According to the article neither the Wachowskis nor Joel Silver were involved at that stage, although the studio would like to get at minimum the blessing of the Wachowskis. In May , it was reported that Chad Stahelski , who worked as stunt coordinator of several films by the Wachowskis, including the Matrix trilogy, claimed that the sisters are involved with the new Matrix film, although he was not sure whether they would be directing it.
The following is a list of crew members who have participated in the making of the Matrix film series. It included all three films, The Animatrix , and six discs of additional material, including the documentary film The Matrix Revisited , the live action footage shot for Enter the Matrix , and a promotional compilation of The Matrix Online.
For this release, The Matrix was remastered under the supervision of the Wachowskis and Bill Pope to improve its picture quality and make its coloring closer to that of its sequels. At the request of the Wachowskis, as they explain in a written statement that accompanies the boxset, each of the three films is accompanied by two audio commentaries, one by philosophers who liked the films, and another by critics who did not, with the intention that viewers use them as reference points to form their own opinion.
It encases the ten discs plus a resin bust of Neo inside an acrylic glass box. The HD DVD release added a picture-in-picture video commentary to the three films and the extras the original standalone DVD releases of the films had. While The Matrix received largely positive reviews, and The Matrix Reloaded received generally positive reviews,   the overall critical response to The Matrix Revolutions was negative. The Matrix films makes numerous references to films and literature, and to historical myths and philosophy including Buddhism , Vedanta , Advaita Hinduism , Christianity , Messianism , Judaism , Gnosticism , existentialism , obscurantism , and nihilism.
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Many references to Jean Baudrillard 's Simulacra and Simulation appear in the first film. Baudrillard himself considered this a misrepresentation,  although Lana Wachowski claims the point the reference was making was misunderstood. Japanese director Mamoru Oshii 's Ghost in the Shell was a strong influence. G , which produced Ghost in the Shell , noted that the anime's high-quality visuals were a strong source of inspiration for the Wachowskis.
He also commented, " I'd imagine that The Matrix is the kind of film that was very difficult to draw up a written proposal for to take to film studios. Comparisons have also been made to Grant Morrison 's comic series The Invisibles ; Morrison believes that the Wachowskis essentially plagiarized his work to create the film. In addition, the similarity of the films' central concept to a device in the long-running series Doctor Who has also been noted.
As in the film, the Matrix of that series introduced in the serial The Deadly Assassin is a massive computer system which one enters using a device connecting to the head, allowing users to see representations of the real world and change its laws of physics; but if killed there, they will die in reality. Matrixism is a new religious movement inspired by the trilogy. Following the Wachowskis' coming out as transgender women some years after the release of the films, the first film and the pill analogy have also been analyzed in the context of the Wachowskis' transgender experiences.
In this case, taking the red pill and living out of the Matrix symbolizes exploring one's own gender identity , starting the transition and coming out as transgender, as opposed to a continued life in the closet. In a summary judgement for the defendants , Judge R. Gary Klausner stated "The basic premises of The Matrix Trilogy and The Immortals are so different that it would be unreasonable to find their plots substantially similar.
In acknowledgment of the strong influence of Japanese anime on the Matrix series, The Animatrix was produced in to coincide with the release of The Matrix Reloaded. This is a collection of nine animated short films intended to further flesh out the concepts, history, characters and setting of the series. The objective of The Animatrix project was to give other writers and directors the opportunity to lend their voices and interpretation to the Matrix universe; the Wachowskis conceived of and oversaw the process, and they wrote four of the segments themselves, although they were given to other directors to execute.
Many of the segments were produced by notable figures from the world of Japanese animation. The first of three video games related to the films, it told a story running parallel to The Matrix Reloaded and featured scenes that were shot during the filming of The Matrix Reloaded and The Matrix Revolutions.
Two more The Matrix video games were released in The Matrix official website also provided several original Adobe Flash -based browser games. The Matrix Comics is a set of comic books and short stories based on the series and written and illustrated by figures from the comics industry; one of the comics was written by the Wachowskis and illustrated by the films' concept artist Geof Darrow. Most of the comics were originally presented for free on the Matrix series' website;  they were later republished, along with some new material, in two printed trade paperback volumes.
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The Matrix official website provided a free screensaver for Microsoft Windows and Mac OS X, which simulates the falling " Matrix digital rain " of the films. From Wikipedia, the free encyclopedia. The Matrix Revisited The Animatrix See also: List of minor characters in the Matrix series. Main article: List of awards and nominations received by The Matrix franchise.
What we were trying to achieve with the story overall was a shift, the same kind of shift that happens for Neo, that Neo goes from being in this sort of cocooned and programmed world, to having to participate in the construction of meaning to his life. And we were like, "Well, can the audience go through the three movies and experience something similar to what the main character experiences?
The second movie is deconstructionist, and it assaults all of the things that you thought to be true in the first movie, and so people get very upset, and they're like "Stop attacking me! I mean, Derrida and Foucault , these people upset us. And then the third movie is the most ambiguous, because it asks you to actually participate in the construction of meaning.
Main article: The Animatrix. Film in the United States portal Australia portal Film portal. The Library of Congress. Retrieved June 29, Den of Geek. Retrieved July 8, Paramount Pictures Acquires The U. February 15, Retrieved March 8, Burbank: Warner Bros. August 1, Archived from the original on March 5, Retrieved April 10, Retrieved March 31, Retrieved March 25, November 14, October 28, Retrieved March 15, February 4, Exclusive ".
March 17, The Guardian.
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Retrieved April 21, Retrieved March 19, Retrieved March 29, Box Office Mojo. Retrieved June 14, Retrieved June 20, Retrieved December 10, October 21, Archived from the original on October 21, Andy Wachowski and Lana Wachowski. Event occurs at Retrieved March 27, — via YouTube. The William Gibson Blog. February 13,