Storage Access API provides a way for embedded cross-site content to check whether it currently has access to browser-based storage, such as cookies, and to request storage access if it doesn't. The API only enables embedded sites to access its own cookies, the same cookies it would have access to when users visited as a top-level site. In this video, I'll explain how it works, when you should use it, and what alternative solutions you should consider. Let's foresee why we need the Storage Access API. Third-party cookies are the main mechanism that enables cross-site tracking, and several major browsers have already placed restrictions on third-party cookies in some way. In 2024, Chrome is planning to phase out support for third-party cookies. We have already introduced new cookie functionality and purpose-built APIs to continue supporting legitimate use cases while preserving user privacy. At the start of January 2024, we will disable third-party cookies for 1% of Chrome users, which means you want to be sure that your site is ready. Third-party cookies also enable many valid use cases, such as managing state-in-embedded content or enabling user sessions across multiple sites. New APIs, like Chips, related website sets, and FedCM, will cover many of these use cases. Still, there are user experiences that rely on unpartitioned cookie access today that are not fully covered by those technologies. For example, authenticated cross-site embeds, such as commenting widgets or subscribed video services, often rely on cookies to authenticate users who are already logged in to their first-party services. When browsers block access to unpartitioned cookies by embedded sites loaded in a third-party context, Storage Access API can be used as a fallback to enable those user experiences. So, how does the Storage Access API work? Storage Access API has methods for sites to check whether they currently have access to unpartitioned cookies and, if not, to request it. Has Storage Access is used to check if the embedded site has access to unpartitioned cookies and other storage in the current context. It returns a promise with a Boolean result, indicating if the document has Storage Access or not. Request Storage Access is used to request access to unpartitioned cookies and storage for an embedded iframe document. It returns a promise that will resolve if access is available. If the request is unsuccessful, the promise will be rejected. There are a few requirements to using Storage Access API in Chrome. First, request Storage Access can only be called from within an iframe. Second, the user must have visited and interacted with the site that owns that iframe as a top-level document, not in an iframe. And third, the user must interact with the iframe before Request Storage Access can be called by, for example, clicking a button or a link. The request will present users with a prompt to either allow or deny access. As with many powerful features like camera or location access, we provide the user with the ability to grant or deny access. You need to consider this in your user flow. Make sure the steps to and reason for granting Storage Access are clear to your users. And of course, make sure you can gracefully deal with them denying the permission as well. Storage Access is an existing web platform API that is also supported in Firefox, Edge, and Safari. Recently, browsers have worked on specification improvements that address security considerations and added features such as permission API support. So if you have already implemented Storage Access API for cross-site cookie access in other browsers, make sure to test it in Chrome as well, which supports the latest version of the API. Storage Access isn't going to be the right solution for every cross-site cookie situation. If your third-party cookie is only being used in a one-to-one embedded context with the top-level site, for example, in third-party chant embeds, map embeds, or third-party payment embeds, consider using CHIPS, which stands for Cookies Having Independent Partition State. Adding the partitioned attribute to your cookies will allow cross-site access with a separate cookie jar used per site. If you're using Storage Access API across a small group of meaningfully linked sites, related website sets can offer a smoother user experience without prompts for sites within a set and additional functionality for the top-level site, not just the iframe, to grant Storage Access. There is also a wider set of purpose-built APIs that may allow you to migrate away from cross-site cookies entirely. For example, your identity provider may support FEDCM to let you handle signing without cookies. To learn more about these APIs, check out our videos and documentation. Now is the time to get ready for the future without cross-site tracking. Audit your use of cookies and plan the actions needed if your site is impacted. Thanks for watching.


What's going on, everyone? Sam here with late in sports cards. We've got two boxes for Joseph S. We've got one, 23, Panini Daunras UFC, and one, 23, Topps Finest Baseball. So good luck here. Joseph, again, no base ship out of this. Everything will ship out of Finest. Irberto, you are not, there's still two ahead of you. There's one Finest, then Imani with a Star Wars, then you with your Daunras Blaster. So probably another, I don't know, 20 minutes or so. You'll go tonight for sure.  Mike Beltran, press-proof Neil Magni. Press-proof is Ricky Simon, purple press-proof Johnny Walker, Habib Base, and Derek Lewis. Got you, got you Irberto. Pull through those. Red Rookie, Matt Brown, press-proof. That is Tatsuro Tyra. Tim Means is gonna be your red to 75. Tim Means. Rosanama Hewnes, crunch time of Alex Pereira and Volpinovsky. Some guy named Barry Sanders' birthday as well. Fair, fair, fair. That guy probably wasn't that good. Yeah, that's nice. Nah, nah, that guy's stunk. That is a press-proof Holly Holm. Nice one there, I could buy a press-proof. No, I did not. Press-proof Holly Holm. Aralovsky, there's Edson Barboza to 99. Mendenuñez, I think there's an Anganu in there. If I do not get your order, we'll go tomorrow. I don't think I'm gonna get to everybody in queue right now. So if you are planning on ordering a personal, I'd wait till tomorrow. Dan Miraglota, what is it? Miraglota, nice one there. Oh yeah, the ref, nice. Authority of the Octagon to 99. This dude's yoked, dude. This guy's so jacked. Several of the refs are. Oh my god. Talia Santos, purple press-proof. Karna Magomedov, that is blue press-proof. Drakkar Klose, Vera the Untouchable. Arnold Allen, authority of the Octagon. Jason Herzog. Jessica Andrade, press-proof. Jack Hermanson, there's Chimov. Khabib, the Untouchable and base. There's, nice. Tito Ortiz, Kevin Costellum. Sean Brady, purple. Marvel's Leon Edwards, Volkanovsky, Chuck Lodell. I know, 2,053 yards and 16 games.  Yeah, that's true.  Terence McKinley, there's a Justin Gaichi, press-proof. Jared Kennear, press-proof. Damon Jackson, purple. Touchable, Cosmov, Chimov. Cyril Gane, nice one there on the Octagon canvas, materials. Heavyweight, Yodong Gong. Sayid, Nurmagomedov. That is Vivian Arujo. Base, Laspakia. Don Russ, UFC at least. That is Daniel Rodriguez. Israel Adesanya, press-proof, nice. That is gonna be Jorge Mazvedal, nice, 2.99. Rats on that. Base and base. All right, that'll do it for UFC, so that will ship. These will not, obviously. 23 Finest, good luck here again, Joseph. Let's see what we got. And Machado to 199, Teal. Refractor, Joey Manessus. Rookie Otto for the national. Nice one there, Joseph. Here, Mitchell. Brett Beatty, O'Neill Cruz. Refractor, Joey Votto. Nolan Gorman and Jay Rodden. Bryce Serang, there's Fernando Tates Jr. Oswald Paraza and Brendan Crawford. Vaughn Grissom, Anthony Volpe, Rookie. The 300 is Shohei. That's one there, Angels. Or not Angels, for Joseph, for the Angels. Gunner Henderson, Rookie, and Contreras. Libertor to Grom, Bonela, Refractor, Rookie. Wander and Tates. Headliners, Juan Soto. Jordon Alvarez and Oswald Cabrera. Go to that and make sure to bring your coffee. Thank you, you're Korean. What'd you say? There's the 250, Jose Ramirez. Blue Teal Vapor. Soto, engulfment. Spencer Stier, Brendan Crawford. Refractor. Correa, Bryce Harper. Vargas, we got an Orange. That's Aaron Judge to 199. Orange Rookie Otto. Michael Harris, four of 25. There we go, dude. Finus has smoked it today. Hell yeah, Joseph. Congrats, buddy. Four of 25. The NL, Rookie of the Year in 2022. Michael Harris, the second. Very nice. Congrats. We'll get that into a mag for you, buddy. Michael Harris, base Rookie and a Nick Perrado. Well, there. Headliners, Adrian Beltra, Shay Langlears and Brian Baer. Bran, Refractor, Oswald, Peraza, J-Rod, and Garrett Mitchell. I know, that's true. Whenever it's an Orange, it's like a huge Rookie. Brett Beatty or it's like a Vat or something. Gunner Henderson, Rookie. Bran Crawford and Nolan Corman. That'll do it for your break there. Joseph, congrats on your hits, man. We'll get them right out to you, buddy.


Can you describe them for us? I am covered with poop all around me. I have poop over my shoulder, poop by me on the computer here. Poop is so important.  Could we suggest more personal hygiene or no?  Thank you for being precise there, Steve.  Poop emojis. That's right. You know, our microbiome metabolizes the food we eat and those small metabolites get into our bloodstream and these metabolites run the chemistry of life. So for the steps that we can no longer perform, if our microbiome metabolites filled in those gaps, then our ancestral mothers had reproductive success and at that moment in our lineage, we exported that gene from our ancestral DNA to a microbiome DNA, which is why having a diverse microbiome is so, so helpful. So we talk a lot about poop in my clinics. We talk a lot about poop in my clinical trials and I ask people, are you pooping rocks, logs, snakes, pudding or tea? If it's pudding or tea, you need less fiber, so cut back on the fiber, cut back on the raw vegetables. If you're pooping snakes, that's perfect. If they get into your pants, you probably have to back off a little bit, because that's socially not very helpful. If you're pooping logs very comfortably and easily, that's okay. If you're pooping rocks, you need more vegetables, more fermented foods, and more fiber. I think I may have shared with you, but if I didn't, I know you're fondness for nine cups of vegetables and when I was writing the Plant Paradox, I said that when you look down in the toilet bowl, you should see a giant anaconda looking back up at you. And my editor said, wait a minute, there was a movie called Anaconda where it's coming out of a toilet bowl and I don't think we want that visual and so we changed it to snake, but I credit you with that passage in my books. Oh, perfect. I'm happy for that. All right.  Well, sugar is really terrible for us all. It feeds the wrong microbiome. Gluten and casein have a lot of cross reactivity with structures in our brain and that drives increased inflammation and molecular mimicry damaging structures in our brain and in our cerebellum. So that's really bad. The third most common food sensitivity is eggs. So that's why I take eggs out and then soy can be a problem, peanuts can be a problem. So in my plan, I have sort of a stage process. So at the level one, we take out gluten, dairy, and eggs and ramp up the vegetables. Then I get progressively more restrictive as people are willing to go on a more intense journey towards health.  Where does that fit in with all these vegetables? Well, I really like to have people get organic meat, grass-fed meat according to what they can afford. I really like them to have liver once to twice a week. If they can find it heart once a week, it would be ideal. Oysters, mussels, that would be marvelous as well. When you get these organ meats, we get more fat soluble vitamins, more coque in that meat. If it's a grass-fed animal, then you're going to get vitamin K2, particularly in that liver. You'll also get retinol. And if we're thinking, well, I can make retinol or vitamin A from the beta carotene, that depends on the genetic variations of the enzymes that convert beta carotene to retinol. And I tell my patients that if they have a autoimmune illness, if they have a cancer or cancer dysplasia illness, that tells me that they're enzymes that involved in making retinol are less efficient. And they in particular would do very well at eating liver. So in my VA, we would teach people how to make green smoothies, how to make cooked greens. And we talk a lot about how to make liver, so it would actually be quite tasty. And by the way, my kids like liver and onions, when my daughter is going to make an exotic meal for friends, she will include liver in that meal. So what do you say to the argument that meats in general are aging and beef, red meats may contribute to cancer? I know you're from the Midwest, and that's where I came from. So if you're meat, if you have a really high protein intake, you may be increasing your mammalian target of rampa myosin. And that is a pro-growth hormone. And so you could be increasing your rates of benign and cancerous tumors, which is one of the reasons I am not a proponent of the carnivore diet, except you're going to have a high mTOR. I would much rather moderate the meat intake. So compared to the paleo diet, the meat intake that I recommend is actually quite a bit lower. So I think meat is complete protein, it's good for us, if you have grass-fed meats. If we're having factory farmed meat, that's a different kind of meat, and it may be more inflammatory. When people are looking at the research that says the TMAO is a problem, and that may increase the atherosclerosis, what they forget is it has to do with the microbiome. And did that meat become harmful because the person was eating a diet high in trans fats and high in sugar, which changed the microbiome remarkably away from the type of microbiome our ancestral mothers and fathers would have had? Yeah, and I think there's some really exciting research that actually was, to the Cleveland Clinic's credit, done partially by them that there's components in a lot of extra virgin olive oil, balsamic vinegar, and red wine that paralyzes the enzyme systems of bacteria to keep them from making TMAO from choline and acetyl-l-carnitine. I think this is an interesting but complicated story. If red meat was a problem, our ancestral mothers and fathers would not have had reproductive success. So what change? It's not just the red meat, it may be what we're feeding our conventional farmers, it's certainly what we're feeding ourselves. For eating our six cups of grains, nine cups of vegetables, we have a very different microbiome. And I think it's an interesting question that we're going to continue to tease out. For a person with MS, or for that matter autoimmune diseases, what sort of foods are good for you? And I think you've alluded to that, but what's your prescription?   The greens have lots of vitamin K and carotenoids, which are critical for retina health, brain health. And there's more and more research coming out that vitamin K is very important in making myelin and very important to brain stem cells that are going to be involved in making the cells that will be involved in repairing the myelin damage and repairing the synapses. So greens incredibly important. The cabbage family, onion family, vegetables boost the detox pathways. They also boost pathways involved in neurotransmitters, mushrooms, boost your natural killer cells, which are good for your immune cell function. And then the deeply pigmented foods are rich in polyphenols and antioxidants. And we have many, many studies showing that the more color you eat, the lower the rates of dementia, lower rates of cancer, lower rates of diabetes, lower rates of mental health problems. And that we have some great studies that have used even just a cup of blueberries, equivalents in randomized controlled trials that show in just 16 weeks or 24 weeks, we can get improvement in thinking. So those are very powerful reasons to ramp up those vegetables. I encourage fermented foods and seaweed as well. Great. Where does olive oil play in your program? Swim in it, drink it. It's really good. There are so many studies that show the benefits of olive oil for cardiovascular health and for brain health as well. I love for people to use olive oil. I prefer that they use it cold because when you heat olive oil, you lose a lot of the antioxidants and you accidentally make some trans fats in the skillet. So I'd much rather people bake roast steam and then pour their olive oil on cold at the end. So lots of olive oil. I'm very keen on ketogenic eating.  Yeah, I think that's something I stress. I take care of a lot of people with the apoE4 mutation. Coconut oil, and I follow their lipids, and coconut oil, particularly in these folks, really makes their small dense lipids go up and they oxidize those small dense LDLs. Yeah, if you're going to go on a ketogenic diet, I have a lot of folks on a ketogenic diet. It's very popular right now and they use lots of butter, lots of bacon, and they're not advising these folks to monitor their lipids. I think that's a disservice. I think you need to know how your lipids are responding. So you can sort out, is olive oil what you're going to use? Do you have to use Pratic Fasting instead? Or are you able to use butter? Are you able to use coconut oil? That's a very important question. So you can start whichever way your culinary tastes take you, but check your lipids, please, and then you can decide which way to continue.


Good morning, John, and the grand tradition of people on the internet assuming that others care about and aim details of their lives, I have decided that today I'm going to give you a tour of the inside of my nightstand, as ultimately this probably says more about me than any home tour or drugstore haul video ever could. First we have my iPad, which I use as a slightly more convenient and vaguely futuristic way to look at tweets.  So now I have to hide it. Okay. Sure. This is just trash. Ooh. An octopus. And a dollar. I got these after my eardrums ruptured during a flight because I had a sinus infection, but I have not had to use them since, so I have no idea if they help. Granny Potty Mouth gave me a business card so big that it is obviously a postcard. There are actually lots of business cards in the drawer. I like how she's just sort of vaguely gesturing at a cloud of pink stickers here. This is a bunch of lens cloths that I got from an optometrist fan on tour. I never cleaned my glasses though because as previously mentioned, I am oblivious to everything. More stickers. Oh my god, I'm only two burritos away from a free burrito. I had no idea this was in here. Oh god, this is the list of people I need to write thank you cards to for helping me with my book in one way or another. Now I'm stressed out just knowing that it exists. I'm sorry. This is actually something that should be in here. It's nice to have a notebook. Here are my notes from my Creator Track keynote at VidCon Australia last year. This is a fart noise machine.  It doesn't even have batteries. Another sticker. I'm pretty sure that this card was supposed to give me access to FX's streaming platform, but I could not be bothered to type in the code. Anybody else want it? Okay, here are some more stickers. Cables, cables, cables, cables, cables, cables. What the frick is this one for? Pens, pens, pens, pens, pens, pens, pens, pens, fancy pen.  I'm a terrible person. I have heard that everybody wants a rock to wind a piece of string around, so I have one of those. Oh my God, Gilbert! You might be thinking right now, that's actually the first enamel pen we found. Don't worry, I keep them all in this Snitchwiches box. Everybody should have a Snitchwiches box for their enamel pens. Look at them all. I don't have a problem. Hey, if you're into enamel pens and want to help support the work this community is doing to decrease maternal mortality in Sierra Leone, the Pin Club is open for just two more days. $12 a month free shipping anywhere in the world. These are the dice I used in Titan's Grave, so I'm glad I had somewhere to put them. I didn't lose them. Look at them. Yeah, memories. Oh great, a cufflink. I definitely eat cufflinks. And this is a shoelace? Just getting to the dregs now, these frickin' things. Does anybody use these? The only thing they do is go in frickin' drawers. Oh, a lone loose puzzle piece. I'm sure that's not a problem. And this frickin' piece of useless metal that my government is too busy being terrible to not get rid of. Cool. Cool, cool. So what did we have? 22 business cards and one business postcard, two flash drives, eight writing utensils, one D22 lip balms and some cold sore medication, a bunch of earbud replacements and other ear things, six miscellaneous peripherals, one recipe for pot roast, a bunch of actual trash, 17 stickers, four cards to daddy, so many cables, $1.60 and like $1,200 of electronics. So now, how's about we make some actual decisions, because I'm not a child. Oh, God, that feels so much better, except for frickin' fine. You go back in, too. Again, it's your last chance to sign up for the DFTBA Bizarre Beasts Pin Pal Club. John, I'll see you on Tuesday.


We were looking at vibrations of a clamped square membrane in two dimensions. It was governed by a linear wave equation, a 2D wave equation. We had started with doing normal modes on it and using the kind of coordinate system. So, this was a Cartesian coordinate system and since the membrane was clamped at all ends, it allowed us to do variable separation. So, we said that the eigen mode can be written in variable separation form as some capital X which is a function of small x into capital Y which is a function of small y. Substituting that, we had obtained an equation of this form. Now, once again I would like to highlight that whenever we do normal mode analysis, it always leads us to an eigenvalue problem. Even this has the structure of an eigenvalue problem. You can readily see this that this can be written in the form because Y is just a function of small y. So, I can write it in the form del square by x square del y square of xy is equal to of xy. So, this operator is a familiar Laplacian operator. So, the Laplacian of xy is equal to lambda times xy. xy is our eigen function A and so grad square of A is equal to lambda of A. So, lambda here is basically our minus omega square by c square. So, you can see that boundary conditions like before will discretize the values of lambda. Only certain values of lambda will allow a non-trivial solution to this equation which satisfies those boundary conditions. So, that will determine our eigen frequencies and in turn will determine our eigen modes of the system. Once again we will write our final answer as a linear superposition of all the eigen functions. Here there will be a, here again there will be a summation from 1 to infinity, but here because there are two eigen functions in two directions there will be a double summation. So, let us work on that. So, you can see that this is an eigenvalue problem and so now we are working on this equation where I have divided the equation in this equation and where I have divided throughout by xy. So, now let us work on this equation. So, this equation can be further written using variable separation arguments. I would like to separate out everything which depends on small x on one side and everything which depends on small y on the other. Now you can see that we have a pure function of small x on the left hand side and a pure function of small y on the right hand side, omega by c is a constant. So, that is not a function of y and so because small x and small y can be varied independently. So, the only way this equality can hold good is if each of these expressions is equal to a constant and it is the same constant. We will be choosing our constant to be a negative constant. I encourage you to think what happens if I choose a positive constant. So, I am choosing a negative. So, this is also what is known as a separation constant. a is a real number and so a square is real and minus of a square is a negative constant. So, we are choosing a negative value of the separation constant. So, now if I equate the first, if I write it like this, this is equal to minus a square and so I get an equation for x and this equation, this is just a linear constant coefficient equation, easy to solve and so a cos a x plus b sin a x. In general, these constants a and b will once again be complex constants because our eigenfunction are in general complex. We will take that into account at the end when we write the answer in terms of real quantities. So, this is my expression for capital X as a function of small x. Now, let us work on the second part. So, I have 1 by y e square y by dy square plus omega by c whole square minus a square is equal to 0 and this can just be written as d square y by dy square plus b square into y is equal to 0, where I have defined b square to be equal to omega by c, it is a new variable. I have just called omega by c whole square minus a square as b square. The square is for convenience. Similarly, the square in the separation constant is also for convenience, otherwise we will have to keep carrying a square root every time. So, again, this is also very easy to solve c cos b y plus d sin b y. So, we have got expressions for capital X and small x. Now, let us go over to the boundary conditions which says that the membrane is clamped at all ends. So, my boundary conditions says eta x is equal to l and all y is equal to 0 for all time. So, maybe I should write because eta is a function of time, so I do not have to write this then. Then, I have eta x y is equal to l at all time is equal to 0, eta x is equal to 0, y at all time is 0, eta x y is equal to 0 at all time is 0. Each of this ensures that one side of the membrane is clamped. Remember that x and y both go from 0 to l. Now, let us work on the lower two boundary conditions because those are at x equal to 0 and y equal to 0. So, if you recall our expression for eta, so the only x part of eta is proportional to this capital X and the y part of eta is proportional to capital Y. This is the only small y dependency and this is the only small x dependency. So, it is equivalent to imposing these boundary conditions only on capital X and capital Y.  I am just writing it again for convenience. So, this would imply, so this condition would imply that x of 0 is 0. So, this would just straight away imply that a is 0. This would imply similarly that y of 0 is 0, this would imply c is 0. This is just telling us that we only are going to get a sin series and that is because of the boundary conditions. But now, just like earlier we had a sin series, we are going to have a sin series now, but it is going to be a two dimensional sin series, a sin along x and another sin along y. So, these two boundary conditions are done. Now, we need to focus on these two, the other two. So, let us work on those. Now, x of l small x equal to l is also 0. This comes from here, the first one of those two and this implies x was already equal to, so, is equal to b sin a l equal to 0. So, this is just going to discretize the value of small a. We are going to set this equal to some m pi, once again m is equal to 1, 2, 3 up to infinity. And then this is just going to tell us that a m is equal to m pi by l. I have attached a subscript to m because a cannot be any value, but only integral multiples of pi over l and so on. The second of those conditions tells us that y of l, when small y is equal to l is 0, this implies d sin b l equal to 0. And this also implies that once again, this will discretize the value of small b and we should not use the same index m. So, I will use another index. So, I will use n and so b n is equal to n pi over l like m, n over also goes from 1, 2, 3 up to infinity. So, let us now write down what are the eigenfunctions that we have found. Our eigenfunctions look like x m of x is along the x part of the eigenfunction is b m sin m pi x over l and y n over x is equal to d n sin n pi y over l. And m and n go from 1, 2, 3 up to what are the allowable. So, this together when they are taken a product, you get the eigenmodes. Now the eigenmode will require a double index m, n in order to indicate it because there are two discrete indexes. What about the eigen frequencies? We have seen earlier that b square was defined as omega by c whole square minus a square. We have seen now from the boundary conditions that b and a are discretized. So, b has a discrete index n, a has a discrete index m and consequently this omega is going to have two discrete indices m and n. So, let me rewrite this again. So, we are going to have b n square is equal to omega m n by c whole square minus a m square. Once again m and n are subject to that restriction. So, you can see that this tells you that omega m n square is equal to c square into a m square plus b n square. And if I write the square root of this, then this is plus minus c, a m was basically m pi by, so this is m pi by l whole square plus n pi by l whole square and this can be written as plus minus. So, there should be a square root here and this can be written as c pi over l m square plus n square to the power half. So, omega m n is equal to plus minus c pi over l m square plus n square to the power half that is my frequency relation. You can see that this generalizes what we already knew. If you substitute n is equal to 0, then you will get omega m is equal to or omega m 0 is equal to plus minus c pi over l into m plus minus c pi over l into m. And that is exactly what we had found earlier for our analysis of the 1D wave equation. There we had written c as a function of c has been had been written as square root t by rho. Here rho would be an aerial density of the membrane. So, once we do this, so here our dispersion relation is written in terms of the wave speed c instead of tension and the density of the membrane. So, now this is our frequency relation or dispersion relation. Like earlier, we can once again write the final answer as a linear combination over all the eigen modes. There will be a double summation now one over m and another over n. So, let us write the final answer. So, the final answer. So, we have found our eigen function now have two indices a m n of x comma y is some b of m into d of n into sin m pi x over l sin n pi x n pi y over l. And these two constants can be combined into a single constant. We will call it e and it will have two indices m n and then the same thing. So, this is a m n and in general, this is a complex constant. So, how do we write the final solution of our 2D wave equation of a membrane which is clamped at all all edges? The general solution is once again it is basically summation m is equal to 1 to infinity.  And then we need to add the complex conjugate of this part of this part. So, you will add a e m n bar and you will add a e to the power minus i omega m n t and the eigen functions will remain the same. So, you can write it like this. And so, if you express it in real notation, it would just reduce to again a double summation over I am just going to write it as a single summation just to as a shorthand and then sin m pi x over l into sin n pi y over l into you would have e m n plus e m n bar into cos omega m n t plus i times e m n plus e bar m n into sin omega m n into t. So, that is my final answer with omega m n being given by the expression. So, now we have added the minus part also. So, I can ignore the minus sign in omega. So, it is just c pi by l m square plus m square to the power half. So, together they give me the most general solution to the clamp-pembrane equation. And you can see once again that this part is a sum of e and its complex conjugate and i times this e minus e bar is also. So, both these are real. So, you can replace e plus e bar and e minus e bar with some constants, but you will have to put some indices on top of that or double index c m n and d m n let us say. And so, once again you we have written it down as a linear combination of the eigen modes. Now, in order to find out these constants, we will have to specify the initial shape of the rectangular membrane. So, if you give it some perturbation eta of x y comma 0, this would be some function let us say let us call it some capital F of x comma y. And then you will have to substitute t equal to 0 in this expression and you will get a double Fourier series. Once again the coefficients of the double Fourier series will be obtained using orthogonality conditions. That is also not enough, you will also have to specify the velocity of the membrane at time t equal to 0 at every point. So, once again eta t at x. So, eta t is del eta by del t. So, I am just taking a partial derivative of this expression with respect to time. And this will give you at time t equal to 0, this will also give you another function of x comma y. This is exactly the same as what we had done earlier, earlier we were doing it for only one space dimension, now we are generalizing to more than one spatial dimension. And this will give you again another expression in which you have to have to have set time equal to 0. The resultant expression, so you are going to get some expression here, another expression here, you can write those expressions yourselves. And then use the orthogonality conditions, you will get two double Fourier series, use the orthogonality conditions to determine the coefficients of both of them. Once you determine the coefficients, you will be able to find out what is the value of emn in this series.  Here also, if you want your modes, if you want your membrane to vibrate in a pure normal mode, you will have to choose one of these, you will have to choose the initial displacement. So, for example, if you want the membrane to vibrate in the 1-1 mode, then you will have to choose an initial displacement which is proportional to sin pi x by L and sin pi y by L. If you set it up that way with 0 velocity everywhere, it will vibrate purely in the 1, 1 mode and that mode will have the frequency given by c pi by L into square root of 2, which is given by this expression when I substitute m is equal to n is equal to 1. There is a countable infinity of modes again and the general answer is expressible as a linear combination of them. Once again, it is not necessary that your membrane should vibrate only in a pure normal mode. You can give it any arbitrary perturbation like f of x, y and if you do that, many normal modes will get excited and then the system will vibrate in a linear superposition of all the normal modes. The resultant motion can look extremely complicated and may not even be periodic, it will still be oscillatory. So, now having completed and gained some experience with solving these kind of equations using the method of normal modes, let us go over to one more coordinate system, which is a circular membrane and we will analyze it in cylindrical coordinates. And we will keep things a little bit simple initially so that we will assume that it is axisymmetric and we will learn about Bessel's equations. This is going to be very useful when we later learn about waves on cylindrical interfaces, when we learn about fluid waves on cylindrical interfaces. So, our next topic is vibrations of a circular membrane. We had looked at until now we have looked at a rectangular membrane, now we are going to look at a circular membrane. This is the center and I am going to use so any point has coordinates r and theta, r is the length of the radius vector up to that point and theta is the angle it makes with the dotted line. Now to simplify things a little bit, I will be making the axisymmetric approximation. This is only an approximation, one does not need to make it, our analysis will become slightly more complicated without this approximation. So, let us so you can see that in the base state this is a circular membrane you can think of it as a tabla membrane or the membrane of a mridangam. It is clamped at all ends, it is entirely clamped at its edges. So, once again we are using the equivalent of the boundary condition we had used earlier. So, it is clamped continuously around. So, the displacement all along the periphery is always 0. So, circular membrane is flat. So, in the base state or equilibrium state the circular membrane is flat and clamped at the edge. This clamping is true at all times even when there is a perturbation on the membrane. So, the equations governing the membrane can once again be easily obtained. So, here I am going to again once again. So, the membrane you can imagine that the membrane displacements is coming out of the plane of the board. So, if I indicate the displacement eta then eta would ideally be a function of r theta and t. And so, we are going to have a wave equation for eta once again those the small amplitude displacements are once again going to be governed by eta and the equation would be a wave equation. If I write the generic wave equation in basis free notation then it would be this and this is the scalar Laplacian. So, all I have to do it to adopt it to this problem is to find out the expression for the Laplacian in cylindrical coordinates. Here there is no z. So, I will only have to worry about derivatives with respect to r and theta. So, that would be 1 by r del by del r of r del by del r plus 1 by r square del square by del theta square. Now, because I make the axis symmetry approximation this term is going to go to 0 that ensures that eta is not a function of theta. So, it simplifies my analysis substantially and so I have eta which is just a function of r and t and so my equation governing eta. So, I just have this term with derivatives with respect to r and if I just open it up it would be. One can derive this equation this is the wave equation governing small amplitude displacements of a membrane where there are no non axisymmetric effects. One can derive this equation from more basic principles by applying Newton's second law of motion to a small part of the membrane. We have avoided that and we have straight away written this equation by writing it using vector notation. So, now let us do let us look at we expect oscillatory solutions here as well and so let us do a normal mode analysis of this problem. So, like usual we set eta of r, t is equal to some eigen mode a which is a function of r only because eta is not a function of theta into e to the power i omega t. So, this is our normal mode. So, this should lead us to an eigen value problem. So, if you substitute this into this equation it would lead us to this equation. Once again you can see that this is an eigen value problem. If I write this as the operator d square by dr square plus 1 by r d by dr operating on a of r is equal to some lambda times a of r. So, you can immediately see that this is an eigen value problem where lambda is minus omega square by c square and this is the linear operator. We will analyze this equation we will find its solutions this is related to the Bessel equation. We will encounter it again when we do waves on fluid interfaces later and we will learn about this equation and what are its solutions and we will solve this and determine the frequency relation. We will also determine the eigen modes of this system and then write the final answer as a linear superposition over the eigen modes.


Welcome to Caltrans LSIT LS exam preparation course. One aid in your preparation for California licensure examinations. A word of caution. Don't use this course as your only preparation. Devise and follow a regular schedule of study which begins months before the test. Work many problems in each area, not just those in this course's workbook, but problems from other sources as well. This course is funded by Caltrans, but you and I owe a profound thanks to others, the courses instructors from the academic community, the private sector, other public agencies, and from Caltrans as well. We wish you well and your study toward becoming a member of California's professional land surveying community. Hello. My name is Don Hunter. I work with Caltrans in the headquarters office of Geometronics. During this session, we'll be covering areas and volumes. We'll begin our discussion with a definition of area. Area is defined as the surface within a set of lines. The area within a triangle is the surface within three sides. The area of a circle is the surface within the circumference. Now, area is expressed in square units, square inches, square feet, square miles, et cetera. One object of land surveys is to determine the area of a parcel. The area may be determined by any of the following methods. One, by plotting boundaries to scale and by plotting the boundaries to scale, the area of a parcel may be found by use of a planimeter, or it may be calculated by dividing the parcel into triangles, quadrilaterals, and computing their areas mathematically. Two, for parcels with irregular or curved boundaries, the area may be determined by the trapezoid rule or Simpson's one-third rule. Three, by calculating the area from coordinates, or meridian distances and parallel distances of the corners of the parcel. By calculating the area by double meridian distance and the latitudes of the sides of the parcel. Now, accuracy. Basically, the accuracy of any of the results of these four methods are in order in which they were presented to you. The most accurate would be by the coordinate method, any of the three which we will be covering later. The following figures and formulas appear in appendix f1 through f3 and g1 through g2 of the Caltrans survey manual. Here we have the area of a circle, pi r squared, r, of course, being the radius of the circle. Now we see a sector of a circle. The formula there is area equals pi r squared times theta over 360. The central angle, or theta, is in degrees and decimal of a degree. A segment, the area is equal to pi r squared theta over 360, just as it was in the sector of a circle, but we add minus r squared sine theta over 2. Here again, the central angle in degrees and decimal of a degree. A polygon, a closed plane figure bounded by straight lines or sides. Here we have two examples of a polygon, one a triangle and one a square with all sides equal. The area for the solution of this type of a figure is 1 quarter n s squared, where n is the number of sides and s is the length of the side, times the cotangent of 180 over n. Here are two more examples of a regular polygon, all sides equal. The same formula. Here we have a triangle, which is a three-sided polygon. The area of a triangle is 1 half base times the height. Now there's a second formula for the area of a triangle as shown in the graphic. This is when we do not have any angles, but we know all of the sides of the triangle, and that is the square root of s times the quantity s minus a times the quantity s minus b times the quantity xs minus c. This formula is used only when the three sides of a triangle are given. Here are two more examples of triangles. Assuming all sides are known, the right triangle on the left would be solved by the formula a equals 1 half base times height, since the base and the height are known sides of the triangle. The triangle on the right would be solved by the formula the square root of s times s minus a times the quantity s minus b times the quantity s minus c because the height h must be calculated, and the information to do so is not given. A parallelogram is a quadrilateral with opposite sides equal and parallel, and the formula for getting the area is base times the height. A rectangle is a parallelogram with 90-degree angles. A parallelogram, which is also a quadrilateral with opposite sides equal and parallel and has 90-degree angles. The next figure is a square which, like a rectangle, has opposite sides equal and parallel with 90-degree angles. But added to that, all sides are of equal length. Our next figure is a trapezoid. The solution for the area of a trapezoid is h times the quantity b sub 1 plus b sub 2 over 2. The b sub 1 plus b sub 2 over 2 will give you the average length of the bases times the height. The next figure is a trapezoid, which is a quadrilateral without any parallel sides. And the formula for computing the area of a trapezoid is the quantity 1 half of dh plus within brackets 1 quarter b times the quantity h sub 1 plus h sub 2. Now that we have reviewed the methods of determining the area of a geometric figure, let's put the formulas to work for us. Here we have an exercise. Find the acreage of parcel A. There are a number of methods we could use to determine the acreage of the parcel. I would like you to turn off the video, refer to the diagram in your workbook, and determine the method you would use to solve this problem. When you have determined the method you would use, turn the video on, and we will review a method I recommend. Did you come up with a method? Let's run through my recommended solution and see how we compare. Here we have a parcel again. Our first step is to circumscribe the parcel with a rectangle, with at least one side of the parcel coincident with a side of the rectangle and a maximum number of angle points of the parcel on the perimeter of the rectangle. The orientation of the rectangle must be such that the length of the size of the rectangle can easily be determined. Step two is to determine the dimensions of the rectangle. Side AD is the length 220 feet plus 280 feet, equaling 500 feet. The side AB is 70 feet plus 225 feet plus 120 feet. The 120 feet is the radius of the circle, giving a total of 415 feet. The area of the rectangle is 500 feet times 415 feet, equaling 207,500 square feet. The next step in the solution is to determine the area of geometric figures between the perimeter of the parcel and the constructed rectangle. The first area we will deal with is the shaded portion of the diagram designated number one. Since this is a right triangle, the base and the height is given. This diagram shows the solution of the problem. If the triangle was not a right triangle, or if we did not know it was, we would have to use the second formula shown in the diagram. The next area to calculate is the shaded area 2, which is a square minus a sector with a radius of 120 feet. Here is the solution to determine the area of area 2. Area 2 equals the area of the square minus the area of the sector. Solving for the square, we have 120 feet times 120 feet for 14,400 feet. For the sector, we have pi r squared times x over 360. x equals degrees and decimal equivalent of minutes and seconds. So therefore, from the given, we know it's a 90 degree delta. The sector would be 3.1416 or pi times 120 squared, the radius, times 90 over 360, which would give us 11,309.76 square feet. Now taking the area of the square, the 14,400 feet, minus 11,309.76 feet, we come up with 3,090.24 square feet for the area of this sector. Area 3 on our diagram is a rectangle designated A, B, C and D. Here is the solution for area 3. Side AB is 160 feet long, which is, we get that from the radius of the circle adjacent to it on the lower side. Side AD equals 415 feet minus 125 feet plus 160 feet or 130 feet. The area of the rectangle would be then 160 feet times 130 feet or 20,800 square feet. Area 4 is another rectangle shown as A, B, C, D on the diagram. And here is the solution of area 4. The side AB is 500 feet minus the quantity 120 feet plus 150 feet plus 160 feet. We can get this right off of the diagram, which would equal 70 feet. Side AD is 415 feet minus the 125 feet or 290 feet and the area would be 70 feet times 290 feet equaling 20,300 square feet. Now the last area is area 5. It is a segment of a circle with a radius of 160 feet. Here we see the solution for area 5 and the square footage of 20,106.24 square feet. We have calculated the areas for each of the five parcels. Now we add them to get a total excess, which is 71,996.48 square feet. Now we can determine the area of parcel A. We take the area of the large rectangle circumscribing the parcel A, which has 207,500 square feet and subtract the total area of parcels 1,2,3,4, and 5, which is an area of 71,996.48 square feet. This leaves a remainder of 135,503.52 square feet. If we divide the remainder by 43,560 square feet, we will arrive at 3.11 acres for parcel A. Next, we will calculate the area of a parcel with an irregular boundary. We will calculate the same area three times by the same basic method, but the last two by a more refined formula. At the end, we'll compare the results. The area formed between the traverse line and the lake will be determined by calculating the areas of the triangles, the trapezoids formed by the traverse line, the offset lines, and the edge of the lake. The area on the other side of the traverse line may be determined by a number of different methods. If the traverse and the offsets are accomplished by a field survey, the areas calculated will be satisfactory for most requirements. If the traverse and the offsets are plotted and scaled without the benefit of a field survey, the accuracy of the areas determined will be limited in their usefulness. This figure shows the stations and offsets. What we have to calculate using this data is, from left to right, a triangle, five trapezoids, and another triangle.  The information on this screen is also in your workbook. After you've completed the problem, turn the video back on and we'll compare notes. Okay, let's compare our results. On the diagram shown, we have the triangle with a square footage of 3,454 square feet and the five trapezoids with 8,287.50 square feet, 5,135.00 square feet, et cetera, et cetera, et cetera, down to the triangle. The last triangle having an area of 1,797.75 square feet for a total of 32,072.75 square feet. Here is a second approach to this same problem using the same area. Offsets at regular intervals. This is called the trapezoidal rule. Offsets may be established at any desired interval, 25, 50, 100 feet, et cetera. The denser the offsets, the more accurate the results. This type of solution would be based on field survey data. The formula as shown on the diagram is the area equals s times the quantity o sub 1 over 2 plus o sub 2 plus o sub 3 plus o sub 4 on until we get to the end of the trapezoids. In this formula, s equals the regular offset interval, o, the offset distance. The formula shown here appears in your workbook along with the stations and offsets for this problem.  After you have completed working the problem, turn me back on and we'll compare results. Thanks for turning me back on. It was getting lonely in here. Well, let's compare results. I have 33,101.50 square feet. And the breakdown is as shown on the diagram. We have one triangle with 1443.75 square feet and the regular trapezoids, using the formula, we have 25 times the quantity 52.5 over 2 plus 88 plus 87 and on and on and ending with 68 feet over 2 and the parentheses. Then we have another trapezoid. Oh, going back to the regular trapezoid, I neglected to say that the total square footage for that formula is 27,206.25 square feet. The second trapezoid and the reason we must calculate it independently is because it has a different base. The difference between the stationing there, you will notice 38 feet. So we have to calculate it separately since it isn't part of the regular trapezoids. And it has an area of 2,641 square feet. The last triangle is 71 feet times 51 feet over 2 or an area of 1,810.50 square feet. Totalling 33,101.50 square feet. The same stations and the same offsets as the previous problem, all of which you found in your workbook. This formula will also be found there with this information solved for the area using Simpson's one-third rule.  When you get done, come back and we'll compare results. Well, how did it go? I got 33,206.66 square feet. Let's review the formula and plug in the values in their proper niches. Refer the formula in your workbook. S3, where S equals the regular interval, or 25C, feet. Inside the bracket, O1 equals 52.5 feet, the offset distance for the first interval at station O plus 55. O2 equals 68 feet, the offset distance for the last interval at station 5 plus O5. Inside the first parentheses, summation of the even offset distances, station O plus 80, the second offset, equals 88 feet, plus station 1 plus 30, the fourth offset, 78.5 feet, plus the offset of the 6th, 8th, 10th, and on through to station 4 plus 80, the 18th offset, which is 46 feet. Inside the second parentheses, summation of the odd offset distances, we start with station 1 plus O5, the third offset that is 87 feet. We don't incorporate the first odd offset because we use that at the beginning of the formula. And we have plus the next station is the fifth station, 1 plus 55 with an offset of 69.5 plus the seventh station at 2 plus O5, which has an offset of 50.5, and on through the 17th station, 4 plus 55, which has an offset of 27 feet. We don't incorporate the 19th also odd offset because this also was used in the second term of the formula. Remember we have two triangles and the odd ball trapezoid to calculate, and add the results of the formula. Let's compare the results of the three methods. Method one was 32,072.75 square feet. Method two, the trapezoidal rule, was 33,101.50 square feet. Method three, Simpson's one-third rule was 33,206.66 square feet. Hypothetically, method three would be the most accurate. Let's move on now to other and more accurate methods of determining area. Determining area by use of coordinates. Here we have a parcel with coordinates at each corner. Using the coordinates, we will determine the length of the sides of various geometric figures around the perimeter of the parcel and by the methods used in the very first example and arrive at the area of the parcel. Step one, circumscribe the parcel with a rectangle. The sides of the rectangle must be due north and south and east and west. The west side of the rectangle must run through the most westerly corner of the parcel in a north-south direction. The north side of the rectangle must run through the most northerly corner of the parcel in an east-west direction. The east side runs through the most easterly corner in a north-south direction and the south side through the most southerly corner in an east-west direction. The second step is to determine the area large rectangle, 1, 2, 3, 4, from the differences in coordinates. The third step is to calculate the area of triangle A, B, A, rectangle small a, b, small b, 1, the triangle large b, small b, large c, et cetera. Around the perimeter of the parcel. Then determine the area of the parcel. Turn off the video and work the problem using the diagram in your handbook and come back when you're finished. I make the area 43,325 square feet. Let's review the process. The area of the rectangle, 1, 2, 3, 4, take the easting of the most westerly corner, point A, which is 1910, from the easting of the most easterly corner, point E, which is 2,260. This equals 350 feet. Next, take the northern of the most southerly corner, point G, which is 1,500, from the northern of the most northerly corner, point C, which is 1,710. This equals 210 feet. Therefore, the area of rectangle, 1, 2, 3, 4, has 350 feet on one side times 210 feet on the other side equaling 73,500 square feet. Next, calculate the area of triangle, large A, large B, small A. Take the northern of A, 1,600, from the northern of B, 1,670. This equals 70 feet, one leg of the right triangle. Then take the easting of A, 1910, from the easting of B, 1945. This equals 35 feet, the second leg of the right triangle. Therefore, the area of the right triangle, large A, large B, small A, is 1,5 times the base times the height, or 1,5 of 70 times 35 for 1,225 square feet. Calculate the balance of the triangles and rectangles using the same method to determine the length of the legs. When you add all of the areas of these triangles and rectangles together, you should get 30,175 square feet. Take 30,175 square feet from the area of the large rectangle, 1,234, which equals 73,500 square feet, and you will have 43,325 square feet, the area of the parcel. The next method we will use to determine the area of the same parcel with the same coordinate values is known as the coordinate method. There's a rule, and it states, multiply the northing of each corner by the difference between the eastings of the following and the preceding corners, always algebraically subtracting the following from the preceding. One half of the algebraic sum of the resultant products is the area. The formula for this method is shown on the screen and also in your workbook. Use this formula to determine the area of the parcel.  That formula is kind of nasty until you've worked it a couple of times. Well, I got 43,325 square feet for the parcel. Let's review the problem and see how I got that answer. You start with any corner, but to keep it simple, as possible, let's start with corner A. From the formula n sub 1 times the quantity e sub n minus e sub 2, let's change that to fit our parcel.  Or the northing of A, 1600, times the quantity, the easting of g 2000 minus the easting of b, 1945, is a plus 55.  Put this in the plus column. Next, n sub 2 times the quantity e sub n minus e sub 3 becomes n sub b times the quantity e sub A minus e sub c, or 1670, the northing of b times the quantity 1910, the easting of A minus 2040, the easting of c. This is a minus 130.  Put this in the minus column. Continue around the parcel to point g, which would be ng times the quantity e sub f minus e sub a. Watch the algebraic signs of the product and enter in the proper column, either plus or minus. When you have found all the products, you will have a column of pluses and a column of minuses. Add each column, then take the smaller from the larger, divide the remainder by 2, and you have your area. Simple, huh? The last method we will discuss to determine area is double meridian distance, or DMD. This method uses the latitude and departures of a calculated traverse that determines the coordinates of the angle points of a closed traverse. Here again, we have some rules. Number one, start DMD calculations from the most westerly point in the travers. Two, the DMD of the first course is equal to the departure of the first course. Three, the DMD of the second course through the next to last course is equal to the DMD of the preceding course plus the departure of the preceding course plus the departure of the course itself. Sounds good to me. Algebraic signs must be considered. Four, the DMD of the last course is equal to the departure of the last course with the opposite sign. Five, DMDs are run clockwise. Hint, when working with DMDs, it is best to set up a form to keep track and see what you're doing. The diagram above shows this form. The first column is your line. Under this heading are listed in a column, the lines of the travers, A, B, C, D, et cetera. The second column is labeled latitude. Under this heading, listed in the column, the latitude for the corresponding line opposite line AB lists plus 70 for the latitude plus because it is headed northerly for line CD minus 30 because it is heading southerly. The third column labeled departure. Under this heading, listed in column, the departure for the corresponding line opposite line AB lists plus 35 for the departure plus because it is heading easterly for line EF minus because it is heading westerly. The fourth column lists DMD. Under this heading, listed in column, the DMD for the corresponding line. The fifth column is square feet or double area. Under this heading, you will have two columns, plus and a minus. Under the appropriate column, plus or minus, lists the double area for the corresponding line. OK, turn off the video. Using the information in your workbook, determine the area of the parcel, then turn us back on for a review. By the DMD method, the area of the parcel is 43,325 square feet. Let's review the procedure. The DMD of line AB is plus 35 by rule number two. The double area is the latitude of AB plus 70 times the DMD plus 35 or plus 2450. The DMD of line BC is plus 35, DMD first line, plus plus 35, departure first line, plus 95, departure of the line itself. This equals plus 165. The double area is the latitude of BC plus 40 times the DMD plus 165 or plus 6600. Proceed through the traverse in the same manner, watching the signs. Then add the pluses of the double areas and all the minuses of the double areas. Take the algebraic difference and divide by 2 and you have your square feet. You will note that all the answers for all three coordinate methods came out the same, 43,325 square feet. So it really doesn't matter which system we use of coordinates. They're all as equally accurate. So we always want to use the simplest one that we can. Now we're going to get into the area of volumes. First we'll define volume. It is the amount of material occupying a certain space measured in cubic units of inches, feet, or yards. Now the geometric figures and volume formulas we'll see these figures and formulas appear in the appendix H-1 and H-2 of the Caltrans Survey Manual.  A right rectangular prism, a parallel opi-d with all angles 90 degrees. For either of these two figures, the volume is calculated by multiplying the base times B times H, where B equals the area of the base and H the height. This diagram shows a prism, a solid whose two ends are parallel, similar and equal, and whose sides are parallelogram. The next diagram shows a prism, a solid whose two ends are parallel, similar and equal, and whose sides are parallelograms. A parallel opi-pad is a prism. My god, I think I finally pronounced it right. The volume of a prism is calculated by taking the area of the base times the height. Here we have a practical application of computing volume. This drawing represents a section of proposed roadway. The centerline of the roadway is shown, and cross sections at two stations, A1 and A2, are also shown. To compute the volume in cubic yards, we would calculate the area of the cross sections in square feet by a method discussed in the area segment of this presentation. The method typically used is the coordinate method. The coordinate method formula given previously can be adapted by substituting offset distances and elevations or cuts and fills for the northings and eastings for all of the points on the section that define the polygon. Add the two cross section areas and divide by two to get the average end area. Next, multiply the distance along the centerline between the two cross sections to get the volume in cubic feet. We then have to divide the cubic feet volume by 27 to get the cubic yard value. Now this represents the amount of material in cubic yards that has to be excavated. That concludes the presentation on areas and volumes. However, I would like to take a moment to talk to you about the LSIT or LS examination, whichever you're taking. I would urge you to solve problems with areas or volumes in them, because they are a very economical way of accruing points. I say economical, meaning the time expended for the amount of points given. The one thing you must know are the formulas and also recognize the problem for what it is and what formulas you have to use to solve it. Above all else, you must be accurate in your calculations because this is the exact science portion of surveys rather than the art. So in any of your endeavors in the LS or LSIT, I wish you the very best of luck.


This is stock number 9312X. We are here at Summit Automotive, your new and used Jeep headquarters in Wisconsin. We are checking out today this 2015 Jeep Grand Cherokee Limited. This vehicle has the 3.6 liter V6 Pentastar motor. From this HD video, you will be able to tell that this Jeep is extremely clean all the way around. Granite crystal is the color. This Jeep has brand-new Michelin Premier 26560R18 tires with the factory painted alloy rims. These tires are brand-new. We put these on during our safety inspection and we put them on all the way around. There are four brand-new tires on this vehicle and they are the Michelin brand. So very nice tires as well. Front bumper is in really nice condition. You can see the hasn't even been used for the screw holes for the plate. Has the projector lamp headlamps, the projector lamp fog lamps. Passenger side rim is in pretty nice shape as well and that tire is brand-new too. As we go down this side of the vehicle, you can see just how clean the body is, how reflective and glass-like that paint is. We take these HD videos. So if you are far away or even if you're close by and you just cannot make the trip down, you can still see the vehicle, hear the vehicle, and have confidence in what you're looking at before you even get here. Very clean down this side. Back rim is in excellent shape as well. Didn't see any scuffs or scrapes on that one. And you can see that these back tires are brand-new as well. Four-wheel disc brakes on the Grand Cherokee. If you have never driven a Grand Cherokee, these things ride amazing. And this one is no different. Back-up sensors. The rear gate is in excellent shape. I didn't see any dents or dings on that. The rear bumper is in really nice shape. No cracks or major scuffs on it. Has the power lift gate. Back storage area is in excellent shape as well. You do get these sun shade right here, which is a really nice feature. So you put that up and what happens is no one can see underneath there. Or they call it a sun shade. If you've got groceries and stuff, keeps the sun off of your groceries. Has a spare tire and jacket. Put this gate down. You press that button. It'll beep at you three times and then it'll come on down. And as we go around to this side of the vehicle just as clean as a passenger side, paints in very nice glossy condition. Very reflective inside.  There are no rips or tears on this seat. Power driver seat with factory lumbar. You can see the floor is very clean. Has memory driver seat, power windows, power locks, and power mirrors. Also has auto headlamps. Quickly we'll check out. Does have the capless fuel fill, which is a really nice feature. You're never going to get gas on your hands again. And as we go inside, you can see that this one has the 7-inch customizable TFT display. So you can see where it has the compass and the temperature. You can also do mileage, miles to empty, all that good stuff in there. And you get a digital speedometer. And you can see that this vehicle has 40,702 miles. Has a heated, leather-wrapped steering wheel. Cruise controls on the right side. Bluetooth audio controls and control. These controls are for that instrument cluster. Has the 5-inch touch screen radio. You can see it has the heated seats. The heated steering wheel button. This is also where your backup camera shows up. Dual climate control. Backup sensors. Start-stop, so you can turn that off. What that is, you come to a stop sign and it'll actually turn the vehicle off if all the conditions are right. You have your Quadra Track 2 system here. So you got your snow, sand, mud, rock, four-wheel drive low. And you also have the neutral button so you could flat-toe this vehicle. Has the keyless entry with the remote start. Entering go system. And then down here you have an Augs USB and SD card jack. Passenger seat is very clean as well. No rips or tears on that. I don't think this vehicle's ever been smoked in. Smells very clean. Has the homelink system and the 911, or no, it does not have the 911, my bad. But it has the rear liftgate and the homelink system and your map lights, of course. We'll take a quick look at the back seats and then we will check out under the hood. Wood grain trim throughout the interior. Very nicely appointed. The back seats are just as clean as the front seats. We don't have any rips or tears back here. Has the latch child safety system for your car seats. Factory floor mats throughout back here. And these back seats are heated. You get two USB ports and then a 115 volt, 150 watt power jack.  And we'll take a quick look under the hood. Power heated mirrors with built in directional signals. Under the hood we have the 3.6 liter Pentastar V6 motor. This motor has been on wards top to an engine list four different times. And this particular vehicle has been gone through our shop. Has had a full safety inspection per the state of Wisconsin requirements. Has a fresh oil and filter change. All the fluids have been checked and topped off. Runs very smooth. Engine bay is very clean. And this vehicle is 100% ready to go mechanically. Four brand new tires. And she's all cleaned up and ready to go. To see more pictures of this one or one of our other 400 new and used cars, trucks, SUVs, minivans, Wranglers, you name it, we got it.  Videos of every single used vehicle that we have all at summitauto.com. Thank you so much for checking out the video. If you like this one, give us a call right now<PHONE_NUMBER>. Ask for one of our sales associates to make this Jeep Ears today. Once again, that number is<PHONE_NUMBER>. Thanks for checking out the video. If you want to check out more, go to www.youtube.com. Slash Summit Auto. Remember to like, subscribe, and share on all those videos and this one. 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