I want to introduce my colleague, Ram Rajagopal, who is a faculty in civil and environmental and electrical engineering. Right? Any other departments that I miss? So far, no. But before we start, I just wanted to just quick, we had a, I hope you enjoyed the conversation with Secretary Schultz yesterday. Wasn't that amazing? And, you know, he has so much of just wealth of experience that he could share. I thought it's always the hit. And I just got back from San Francisco, I just drove back. There's a big, as you know, global climate and action summit going on that Governor Brown is holding. And he just signed something called SB 100. Has anyone heard of SB 100? It's the, it's a very coded language, but it is the state Senate bill to go to 100% carbon free energy in California by 2045. Okay. And this is now the law of the land. And amazing. Of course, we don't exactly know how to get there. And which is why this topic is so important out here. Actually, let me correct myself is 100% carbon free electricity, not energy, carbon free electricity for this. And I just, that's going on. I was at a gathering. How many of you have heard the giving pledge? Okay, giving pledge is a group of billionaires and hundreds, multimillionaires who have decided to give away half their wealth for philanthropy, started by Bill Gates and others. And they had a gathering out here to convince them if there's any issue that is important, it is the energy climate nexus is probably the most important because it affects in a food effects migration affects everything that we can think of. So, so this is going on in parallel. And I told him, sorry, as, as important that is, I got to come here. Because I got some students waiting out here. And so here we are in the with a backdrop of 100% carbon free electricity. We'll talk about the grid. And I'm going to ask Rom to give a sort of introduction of what's going on with electricity world. What are we doing out here, Rom? Yeah, so amazing to hear the changes that are happening right now. It's a big opportunity. I think what we are seeing in the grid is a transformation where we are adopting more renewables, which causes intermittency and uncertainty into the usual operations of the grid. And that starts to require us to have new strategies. So the traditional paradigm of the grid has been generation follow load. And that means whenever there is a variability in generation, there has to be other generation to compensate. If you want to do 100% renewables or carbon free in that manner, you would have to have enough backup power to absorb any variability of that. I think the first challenge that you're facing is regarding how to use the all the other resources in the grid from transmission level storage to the ability to control the transmission system all the way down to the distribution network customers and all other assets in order to balance the system better. And I think that also opens up questions around markets, policies, business models, and various other things. Inspired by this, Arun and a group of faculty here have established a program called Bits and Watts, which is to address this question of how to reduce the cost of operating this system overall when we are starting to absorb more renewables and storage and technology like that. That's right. I mean, the goal is to decarbonize the grid because if you don't, we won't be decarbonized at all because the transportation is moving to the electricity as well. So we are seeing some massive changes going on. And I think in the next 20 to 30 years, we are going to see a very different energy system. There are countries that are announcing that they're going kind of all EV or no gasoline car sales after 2030, 2045, you take the number and it's going to happen. So as a result, and they all rely on a grid. And the grid, as Ram pointed out, is going to have volatility at both ends. On one end is the solar and wind integration because they're just the cheapest.  And economically, that's the most affordable thing to do. On the other hand, you have variability on the edge of the grid, whether it is EVs, big loads, EVs are going to be big loads. We will have networked thermostats that we can somehow, someone can remotely, you can sit out here, open up an app, can change your load. And if someone does it for you, and if they 100,000 homes go up one degree Celsius in the set point, and thereby the load goes down suddenly, it can destabilize the distribution network. So suddenly you find volatility at both ends of the grid. And how to manage that is a non-trivial aspect. Very important. And Arun, maybe it's worth if you share a little bit of your experience of how these changes are happening, not just in the United States. That's right. Yeah, this is happening worldwide. It is a problem, not just the US grid, but Europe and India and China, etc. And it's even faster there. It's faster. That's right. And infrastructure is different, and we need solutions that can be customized, tailor made in all these different places around the world. This is an enormous challenge. And we are still living in the architecture of the Tesla Edison grid. That's where we are. And it was never designed for all of these variables. That's right. So why is the name of our initiative bits and watts? Because in Tesla and Edison's time, the only mechanism of communicating between load and generation was frequency. That was the only. So if the load goes up, the frequency goes down. And so someone turns on the generator up so that the frequency comes back to 60 Hertz in the United States, 50 Hertz in other parts of the world. That's how the communication happened. That's how the grid is run today. After 120 years, it's still run the same way. And so we are saying that, well, maybe we could do better, not to say that we'll throw away the whole system. But the idea of bits is data. We all here, I'm sure you've heard about something called IoT, right? Internet of Things. Well, in this case, we will have IoT, but these IoTs will have to satisfy the laws of physics. Whereas your Fitbit does not have to satisfy the laws of physics, at least the electricity physics. So this is the challenge that we have. How do you take data from number of sources around the grid with this transformation system, substation, your home, everything else, and how do you coordinate in a way that can rewire the grid in a virtual way and still manage the volatility at both ends? And this is not just a technical problem. It is a market. It's a pricing issue. It's a business model. Someone has to make money. Otherwise it won't last. Because the whole design of the whole thing was made assuming the grid as today is what's operating. And all the other elements were designed on top of it. If you change that, now we have to rethink everything. And I think it's a very exciting time. Very exciting. If you're an engineer and you're passionate about this, it's super fun. And the corporations and bits and lots from all over the world, and they want to come and join. They more want to come and join. And I hope you guys join as well if you're interested in this topic. But maybe you could give a few examples of projects that are going on, the educations. Yeah. So first of all, the first principle we had in bits and lots was to have a comprehensive education. So we have classes ranging from classes in the business school to classes in the school of engineering and so on. So my course as an example is a course that is called Modern Power Engineering. And it is not your traditional plain vanilla power systems class. Because what we learned with you guys from the feedback of students, from the feedback of industry, from the feedback of faculty is we need students who are prepared to piece the systems together and have a holistic understanding of how the grid works, how to model this whole thing and be able to simulate things in the computer, work with data and so on. Not just your traditional power calculations and wire sizing, transformer sizing. I think those people are very well aware of how to do. And we are now going into this new world where as an engineer in this place, you need to know mechanical engineering. You need to know economics. You need to know civil infrastructure engineering. You need to know, of course, energy systems engineering. And you need to have this idea of how these things piece with each other. And if I change this, how it affects the rest. You can't really change one thing and buy yourself solve the problem. I think that that's the big challenge and why it is also an exciting challenge if you are in the side of learning. You can learn so many things. That's on the education side. On the research side, we started. The first concept we had was to bring machine learning and AI to grid research. For those who took the master class, you saw some examples of that. Arun also gave an example of deep solar, how you can apply AI. Maybe Arun, you can give some examples of what you did at Google as well. Sure. When I was at Google, we asked the question that, okay, so Google, if you go to the data center, it uses megawatts of power. And there's some amazing technology internally within Google on managing electrical power. It's a grid in itself. And in fact, high voltage grid inside the data center because it reduces losses then and it's cheaper. And it enables computing. And we asked the question, okay, can we flip that around? And can we have computing, the most modern network distributed computing enable the grid? So flip that around. And especially with a view that if you're looking at those one and a half to two billion people who do not have access to the Tesla Edison grid, you still want to string them along and do you want to leapfrog them from the 19th century to the 20th century or 19th century to the 21st century? And that's the question we asked. And we had a whole program within Google called the bug project, the bottom-up grid. Instead of bringing top down, now that solar and storage and all are so cheap, can we develop microgrid solutions? And can we do it in such a way that a 10-year-old kid can put the grid together like an extension cord, which is how it happens in many parts of the world where there is no grid is like they keep, you know, in a daisy chaining and wires and soon enough, but you got to make it safe. You got to make it automated. You know, there is no there's no half automation is there's like there's no half pregnancy. You know, there is no half automation.  And people talk about smart grid, but frankly today the smart grid concept is not that smart. We only have meters. The meters are producing data. We don't do much with the data and we don't use it enough for the control of the grid, which is what a true smart grid ought to be. And that's what bits and watts is all about. That was the Google project and some of the concepts and ideas have come out here in the bits and watts project as well. That's where we started. I think it's also a way of starting to see how to use these information technology tools like AI, like machine learning, like networking for some societal good. And energy is an area that can have an enormous impact. You have been seeing this this whole week. This was one of the reasons why I myself, you know, I come from Brazil. Arun comes from India. I know in these two countries lights flicker. I thought it doesn't happen here, but I was sometimes there are no lights. That's right.  The lights flicker a little bit. So, you know, for me, this was an exciting topic, but it always seemed why would the kind of tools and solutions that I love, would they apply to this? And I think it's a resounding yes. So that was an inspiration. That's where we started. And I think the vision is to go well beyond that. It is to start to build the system's view into a technology, into kind of a platform for the grid. And being at Stanford, you're not a company you don't have to necessarily find an immediate profit motive or you're not going to have all the participants of the grid as your antagonists. I think that is the other enormous advantage you have here. It can be in a sandbox. That's right. So before we open it up for question, let me just say one thing that we are now thinking of launching. And we like to see if there are people interested in this. The challenge of EV charging is a new challenge. There is no history to look back and look at past data to see how to design and operate EV challenge, right? EV charging is variable in time and space. Renewable generation is variable in time. We know the weather spaces. In between, there's a grid. Sometimes there'll be congestion on the grid. We won't be able to charge EV. And when you have electric vehicles that are penetration of only a few percent, that's an easy problem to solve. If you're at 30, 40, 50 percent penetration of EVs and you don't have the charge, it's like, you know, your phone is dying out and you're looking for the outlet. And that's not a good place to be. And we may run into that situation. So one of the things that we are looking at right now is a project that we're designing along with the corporations, the private sector. And they need help because this is an intersectorial issue. You've got electricity on one side, you've got automobiles, you've got service companies like Uber and Lyft and others. And in other countries, there's DD, there's all our taxis, others, right? The old mobility service providers. We're coming from the service side and asking the question, what is the scalable architecture? The backend architecture that you need. A to design so that you have enough capacity to provide the services and to operate it so that if I'm driving my EV, the car should tell me that here's a cheap charging station out here. Do it now. It's next 10 miles. If you want to do green charging or you want to do fast charging or do you want to do cheap charging? Those are the options that we may anticipate. But this open platform of a scalable architecture with open APIs and protocols has not yet been developed. And it affects everything again. Because if I have 10 cars charging behind a transformer, that transformer starts to heat up. It's the hottest day of the summer. Suddenly now the transformer has to talk to the chargers and decide in a different operating mode or things will just break down. And if you want fast charging, the system should tell you not to go there. Everybody wants fast charging. So that's the kind of coordination we don't have today. And that's the framework that we're putting together. We'll be engaging with obviously the mobility service companies, the auto companies and the electricity companies, which have not talked together. They don't have a common language. One is regulated. Electricity is regulated. Auto is unregulated. So the language has to be common. And auto industry is driven by convenience. That's for sure. They can't, I mean, they never imagined that fuel or electricity is a constraint for them. So we are going to have lots of discussions with the companies in that sector. This is global. This is not just the United States. And then we'll be putting a research project together with a group of small group of faculty and students to develop their architecture, to develop the hardware software, to implement it first on campus. Okay.  So we'll be eating our own dog food. We may, we may run a pilot with Uber or something. So that's, that's what we're thinking. And if you're interested, you know, and let us know. I think this is a chance to go beyond, you know, math and optimization. I love those topics. It's very elegant to do things in math lab in the computer, but getting your hands dirty and seeing your stuff work, that's what makes a real impact. And this is not just science engineering. This is, you'll have pricing, markets, business models on top of that. Usability. That's right. And so that architecture has to be able to get these layers properly, correctly done. And if you have not done it well, your friends are going to be charging here on campus, they will feel the effects. So maybe we just open it up for questions. We didn't have any, by the way, I think you're getting a lot of slides. We decided not to put any slides for you guys together and just have a conversation with you. Any questions? Yeah. I don't think you mentioned it, but how do you plan on incorporating security into these though? Because I mean, if you're planning on making a more land AI data, it seems to me that it's more vulnerable to cyber attack than all grid is already. Great question. You want to take a shot? Yeah. I think there's a two pronged approach. Arun has been building kind of an effort to bring the industry around security to grid, which is, you can imagine it's a huge challenge because everything is challenged with security and people thought the grid was safe. But we are now saying, look, no, it's not the case. The second one is internally at Stanford. We have the award winning team. It's a team of students who go and win the hacking competition where they're supposed to hack corporate systems and then defend corporate systems. I think they place first on the hacking, second on the defending or something like that. And they came and saw the lab that we put together for you guys. And they were just so excited because they said, we just see so many opportunities to hack. It's amazing. It's like carnival. I can just do this all day. So through the university, I think we want to find students who have this passion for security and working with some faculty in the area and with this group from computer science to figure out this question. It's not a simple question. You ask industry, some guys will tell you it's solved until the next time things are hacked and then it's not solved. It's a spy versus a spy. How many of you read the mad comic book spy? So it's going to go on. And I think the topic of cyber is so important and you cannot add a security protocol to a communication link and say, oh, we are all safe. It is not. We've got to think of it like a system and we've got to have cyber in mind when you design the whole system. And for the EV infrastructure, charging infrastructure, I think that's an opportunity to think of it as a system for all the attack envelope, et cetera, sort of looking at that and figuring it out. You know, if you have a system that's operating at its limit, maybe I add five more cars for charging. I'm not even hacking in the traditional sense. I can bring down some portion of my local grid and all of these kind of things have to be figured out. I think there's a lot. We don't have machine-to-machine two-step authentication. So we have many, when you put your ATM, they ask you for your number and you put your number. That's a two-step authentication. We don't have machine-to-machine two-step authentication. So you're trying to automate a system. You have to figure that out. So those are many, many challenges involved. And there's a good question. I think over there and we'll come over here and over there. So, kind of regarding the electric vehicle, you just mentioned earlier, I was curious because you mentioned the market. I'm thinking that if we have a period of 40-market penetration, hopefully that means that batteries provide energy to the grid from the car. And so I'm curious about what you guys thought about the liability of that idea, that kind of more shared approach to alleviating any sort of penetration. Great question. Yeah, I think when we think about these problems, we do think about the possibility of two-way charging and discharging. And we should point out for 10 cars, 20 cars in the Navy parking lot, people have shown all of this is possible. But those ideas don't really scale out for tens of thousands of cars. And one lesson I learned through the grid and through my career is these kind of scaling problems. Sometimes you only learn the actual questions you need to answer as you are doing the problem. You will start to discover many of these issues. So in an idealized world, of course, you would love to have this two-way charging, discharging. But figuring out what is going to be the bottleneck for that? Is it consumer acceptance? Is it some technology issue? Is it the fact that maybe in the grid, things are much more balanced just by the charging side and so on? I think the grid, my intuition is that the two-way power flow is more difficult problem than the one way because the grid, the distribution network was never designed for two-way. And so this is a major issue. If you are to go that way, by the way, in the United States, none of the EVs are bidirectional. The same Nissan Leaf that is unidirectional over here for charging is bidirectional in Japan because of regulation. It's mandated. And they just have to flip a switch, frankly. And that's because they had Fukushima and they needed power. And so they decided that's what they're going to do. And if you look at a 25 kilowatt-hour battery which a Tesla, which a Nissan Leaf has, roughly 25 kilowatt-hours, you can run your home the whole day. The whole day can be run on a single battery charge because your home is typically about a kilowatt-hour average and you've got 24 hours, that's roughly a day. Yeah, I think there was over here, then there was over there, and then I'll come over there and then do that. Yeah. Yeah, now when it comes to all these devices in the time and space, there'd be an argument that just like IoT, for another from a small size to a big size, even talking about a gigantic computational computing power over here. Now there comes a question of where does that computing is happening? Is it like happening at a centralized place or at some age or age? I think a lot of that is going to be, it'll be a combination of, and IoT is basically in a colloquial jargon for networked embedded systems. There'll be some embedding of computing in locally that will provide the stability of the grid, etc. There'll be all network to the cloud and there'll be where the past data will be stored, which will be analyzed and actually acted upon to send signals down to the grid and thereby optimize this thing. That's the vision that we have. Yeah, I think it's intelligence partitioning, but I think what we are learning is even the scaling of the cloud and all of these issues are so important that I think Google is part of bits and watts and because they have to figure out can this kind of system support. And the partitioning will be based on economics. How much storage do you want over there? How much computing power do you want over there? As opposed to the cloud, which is really cheap right now. So that's the, yeah. So when smart charging or discharging electric vehicle fleet for visibility or new authentication, has there been a research at Stanford in designing incentive structures for consumers to accept having their car charged at a different place or client? Yeah, there has been some small scale efforts on research. I think the research typically starts with some model of how users value their charging. And in my opinion, because I myself am author of some research like that, I think the weakness of that is we don't really understand that. And part of why we want to build this test bed is to kind of learn about these incentive structures and start to design maybe something completely new. Maybe you don't charge, you don't pay each time you go. Maybe it's a flat rate or something, right? That's right. Today the electricity system is run based on dollars per kilowatt hour. Or maybe if you enable your car to be two-way, you're paid for that. But dollars per kilowatt hour is a pretty old way of thinking about it, because if you're doing fast charging, you should be playing dollars per kilowatt. So I think there are lots of very interesting pricing models that will come out of this. But it's a great question because the markets and the business models and all will have to be overlaid on platform of some kind. And then the consumer acceptance, it's another issue. There's a question over there and then over here. And we are then running out of time. Yeah. So that here is a really strong way to do something. So is your question about why not invest on a stronger network? And I think Arun, when you were in RPA, when you created RPA, I think this was one of the issues you looked up deeply.  It's at the end of the day, it comes back to the question of what is economics? Okay. So you can add a lot of resources to the grid and make it very strong and resilient and all that. But someone has to pay for it. And you don't want to over design too much. And I think that's the question is that's the tension between how much technology to make it stronger and how much cost. And I think that's the optimization that has to be played out within a regulatory framework. The Chinese regulatory framework is very different from the US one. Even within US, it's different. You know, as how many people are in the law school, who just as Brandeis, you know, who just as Brandeis is in a very famous saying, the United States, we have a laboratory, the states, we have 50 laboratories of democracy going on. And that's just as Brandeis, right? And so in the United States, the electricity system is not quite 50, but it's like 30 or so different experiments that are going on right now. And there's different regulatory frameworks. And I think the solutions will have to be cast in economic regulatory as well as technology. There's no one solution. Installed in a city, I think plenty of different scales. For example, the whole of Netherlands, there's been a very large scale effort to support EVs and add a lot of these resources in homes. I think in the US, we have had efforts supported by the state in here in California itself. Campuses, UC San Diego, that's right. Is a wonderful test bed. Navy and Army campuses as well. So there is plenty of examples. And I would also add to that, that Stanford campus, our energy systems folks that run the energy system on our campus are extremely open. This is not common. Are extremely open for students to actually add things to it and use this as a test bed. In fact, one of the students that got the bit that was fellowship is doing exactly that. That's right. He figured out how to control better the system and actually save money for campus. So your response to demand response signals and using machine learning and all that. And he's saving money for campus. And if you can do that, the campus will be very happy. I think you had a question. And I think, okay, how much time do we have? Four minutes. Okay, quick question. You've touched on this a few times. But so this structure of the past for the energy system of the past and coming to the future, sort of like this system of the past is the handle of resources of the future. But that it wasn't built for it. And the same is true of the markets and the policies that you need. So I'm wondering to what extent it's not something about things like, I said it's not likely to quite quickly start over. It's all kind of incremental change and how that's a challenge for the resources. We have a former commissioner on our team. Yeah, you can't engage more than that. I think that's right. But yeah, no, it's absolutely. So we deal with Cal ISO, not just Cal ISO, but others around the world, bread operators. And you're absolutely right that we need new market design to be able to manage. I mean, the markets essentially try to satisfy the laws of physics through markets. It cannot do everything, but it tries to do as much as possible, load balance and load generation balance itself. But with new kinds of things that are coming on board, we need new markets. We don't have a futures market. See, here's the challenge. And are you in the business school? So here's the challenge. We make energy investments that last for 60 to 70 years. For example, if you decide to put a nuclear plant, that's going to last for 60 years, maybe longer. But it has to be paid back in the markets of today that has a maximum horizon, the capacity markets over three, four years horizon. We don't have a futures market that takes care of long term investments in the energy sector. And we are trying to make decisions on our energy system on a day head market or a capacity market for years market when these investments last for 67 years.  So if you guys can innovate on markets, that'll be true. And of course, there's faculty even in bits and lots of our looking at that. There are projects around that. And the last question someone had, okay, there you go. Yeah. Obviously, looking in the distributed storage systems to be able to help regulate the specific frequency and voltage and the intermedancy of loads. I think that's where Arun and I have a project. Called PowerNet, which is about that. Figuring out if we have this behind the meter storage and solar and so on, and you need to regulate them to not just control voltage, but also even provide services back to the grid. What do you do? Yes. So we are doing that in the process of doing that. One of the lessons learned has been, well, we are successful at doing that. But if you want to scale to something like EV where 40% of millions of cars are now in a system, we have to create something new. And it's of a different scale of challenge. I mean, batteries is a stationary object. It is pingable and it doesn't move anywhere. It doesn't have desires of its own. It's a way easier to deal with than cars. I think one thing that has always made me think long and hard about all these problems is that sometimes what you think is an answer, then there's kind of another problem, another innovation that is in place that one creates technology problems. But then you go and try these solutions. So we went and are trying to now recruit homes, real homes in Fremont to protect them. To participate in PowerNet. And the kind of questions residential users have about this technology, if you put a control in the battery, of course, will it blow up? Will it, will my house lights flicker? Will you turn off? Will we shut me out of my garage? And what is this doing? Why is it good for me? And it is an enormous challenge to even think about the problem holistically. I think you need to be in a place like Stanford where we are creating the whole energy systems program from the ground up. It's totally new. It's not behold to any of these traditional ideas and traditional ways these things were seen even at the university level, even at the level of you getting a degree and a PhD. It's very different. I don't know, Aaron, don't you find that? Absolutely. Yeah, this is a, the beauty about Stanford and it is represented in this crowd out here is that we have people from, you know, one of the best business schools, law school, engineering school, sciences, humanities, education. I mean, all the schools are fantastic. And to be able to put that intellectual horse part together, to look at the problem holistically is something that I think is fairly unique out here. Otherwise, you may have a great power systems program, but you don't have the market guys. You have a fantastic program in markets, but you don't have the power electronics of our systems people. This is something that we try to do that because that's what the industry or all our stakeholders, that's what they want. They are not monolithic, right? They have to have solutions that all combine and reinforce each other. And that's what we're trying to do. I think it's super exciting and it's a place you can have, you know, a person like Arun and his enormous experience with this real world side. So this is another change in perspective you may want to make as well. I think impact in energy through science and engineering, you know, having that practical impact is possible. We are trying to build up the means to help you do that. And this is a place we encourage you to do that. It's not just about writing papers, which we know you guys are brilliant. I think most of our ideas and bits and lots came from students. Is that fair to say? So, you know, we know you guys are able to do that, but we are challenging you to now take it up another notch because the time is not 15, 20, 30 years from now. When they say 2045, the solution has to be ready in the next five to 10 years because you need to test this thing and scale it up and so on. And build, you know, the whole system out. So it's not something you can wait and somebody else is going to do and so on. By the way, the governor from Washington, state of Washington, James Lee, he was there and when he heard the governor of California, we had a dinner all together. The governor of California said by 2045, he piped up and said, Washington will do it at 2044. So there's a race going on between the state of Washington and the state of California, which is a wonderful thing to have. Maybe we can automate the whole campus energy system and all the EV charging given in the next 10 years or five years. On that note, I think we run out of time. Yes. It was a terrific discussion and, you know, and join us in what you do.


Hello and welcome to the session, let us understand the following question which says, a spiral is made up of consecutive semicircles with centers alternatively at A and B, starting with center at A of radii, 0.5 centimeter, 1 centimeter, 1.5 centimeter, 2 centimeter and so on, as shown in figure 5.4. What is the total length of such a spiral made up of 13 consecutive semicircles? Take pi is equal to 22 by 7. Here is the given figure. With center A and radius 0.5 centimeter, we have a semicircle of length L1 and again with center B and radius 1 centimeter, we have a semicircle of length L2 and similarly, we have a semicircle of length L3, L4, so on till L13 because it is given to us that spiral is made up of 13 consecutive semicircles and these spiral is made up of radii 0.5 centimeter, 1 centimeter, 1.5 centimeter, so on. Now let us write the solution. Let L1, L2, L3, so on till L13 be the lengths or circumferences of semicircles of radii. R1 is equal to 0.5 centimeter, R2 is equal to 1 centimeter, R3 is equal to 1.5 centimeter, so on respectively. Then L1 is equal to pi R1. L1 is the circumference of the semicircle, so circumference of the semicircle is equal to pi R, where R is the radius of the semicircle. Now this is equal to pi multiplied by R1 is equal to 0.5 centimeter, which is equal to pi multiplied by 5 by 10. This gets cut by 2 which is equal to pi by 2 centimeter. Now similarly L2 is equal to pi R2 which is equal to pi multiplied by R2 is equal to 1 centimeter which is equal to 2 multiplied by pi by 2 which is equal to 2 pi by 2 centimeter and L3 is equal to pi R3 which is equal to pi multiplied by R3 is equal to 1.5 centimeter which is equal to pi multiplied by 1.5. This decimal guess goes off and here we get 10. This gets cut off 5 3s are 15 and 5 2s are 10, so it is equal to 3 pi by 2 centimeter. Similarly L4 is equal to 4 pi by 2, L5 is equal to 5 pi by 2 and so on. Therefore L13 is equal to pi R13 which is equal to pi multiplied by 13 by 2 which is equal to 13 pi by 2 centimeter. Now total length of the spiral is equal to L1 plus L2 plus L3 so on till L13 that is the sum of all the circumference which is equal to pi by 2 plus 2 pi by 2 plus 3 pi by 2 so on till 13 pi by 2 centimeter. Taking pi by 2 common so it is equal to pi by 2 multiplied by 1 plus 2 plus so on till 13 centimeter which is equal to pi by 2 multiplied by n by 2 multiplied by a plus L centimeter where S L that is the sum of first n terms is equal to n by 2 a plus l where n is the number of terms a is the first term and l is the last term. So it is equal to pi by 2 multiplied by n is equal to 13 by 2 multiplied by a is equal to 1 plus l is equal to last term which is again 13 centimeter which is equal to pi by 2 multiplied by 13 by 2 multiplied by 14 centimeter. Now 14 gets cancelled by 2 and we get here 7 so it is equal to pi by 2 multiplied by 13 multiplied by 7 centimeter. Given to us this pi is equal to 22 by 7 so substituting the value of pi we get here 22 by 7 multiplied by 2 multiplied by 13 multiplied by 7 centimeter now here 7 gets cut off and 2 gets cut off with 22 so we get here 11 which is equal to 11 multiplied by 13 which is equal to 143 centimeter. Hence required total length is equal to 143 centimeter. I hope you understood the question that's all for the session bye and have a nice day.


The only thing that brings me back to this day yet is diesel fuel smells. I can look at the pictures and it doesn't bother me. But the diesel fuel takes me back. I reported on board the Christian via helicopter out of Saigon. It was an interesting trip. They flew over one area where they somebody seen something and they decided they ran open fire with their machine guns. Okay, take me home now. The ship was ARL 38 and its primary job was to repair the riverboat. They would pull up through our barges, would be lifted up out of the water, put on frames on the barges and be worked on there for repair. We've seen a lot of the same guys out of the riverboat that come in for repairs and go back up the rivers.  We still did watches for our shifts and rotated through the day and the night. We either carried a 45, a M16 or a shotgun, but you also had to throw, I think, six percussion grenades every hour. The hard part about that was when you had a lot of boats tied up alongside the barges. You always had to call over each ship with very little lighting, carrying grenades and then toss them off the sides. You didn't have a schedule tour. It just worked because there was really nothing else to do. You could spend 12 hours working or longer if you wanted to. In the evening there was always a movie on board, what we called the movie deck. Spent some time reading, writing. Everybody would exchange tapes and record large tapes. Later in my tour we had a new recreation officer and he got us some fishing equipment, so I spent a lot of time fishing. You had to kind of find a way to entertain yourself because there wasn't a lot. First night we were up the rivers. We got in late afternoons at Anchors. It didn't seem like there was anything different than an oil station. Not except that the shorelines were real, real close. I mean, to many of us it seemed like you could almost throw a rocket across the rivers. But we anchored and everybody went about their duties. They had a boat on the barge already to go to work. That night around 10 o'clock we were hit with a mine. Somebody came in and had asked me to go get some batteries for them. And as I went back into the storeroom, it was when we were hit. The mine I actually watched it come through. It came through about 10 to 12 feet in front of me. They bothered to tell them that I got the word. That's enough. Go to General Quarters. It's a reaction thing. My biggest fear when I ran up to my battle station, the whole top deck of the starboard side was lined with oil. We'd just gotten resupplied. And I'm thinking if somebody hits this whole ship, it's gone. I mean, it's amazing.  You have to call it good luck because it went through Sick Bay. There was nobody in Sick Bay at the time. That movie that was shown, The Good, the Bad, the Ugly, was a well-known movie and there was a lot of people wanting to see it. There really wasn't anybody in the area where the explosion hit. It went underneath the officers' birthing areas and nobody was injured up there. It went through halfway through the ship. There was just nobody there. It was just very, very fortunate. I never really thought about when you keep doing all these trainings that that day would ever happen and you wouldn't think about it anymore. You knew what side of the ship to go up on. You knew where to go to report. You knew what you had to do when you got there. It happened pretty quick and everybody did exactly what they had to do.


After over three hours of deliberations behind closed doors, the Plateau State Security Council rose from its emergency meeting with the directorate from Governor Simon Lalong to the security agencies in the state to take full control of all vulnerable areas in Bassa and Riumu local government areas and Kaledi attackers that have perpetuated mayhem within the areas. Briefly newsmen after the meeting, Commissioner of Police Edward Egbuka said the governor after receiving full briefs from the various security organs in the state and as an exhaustive review of the situation, gave marching orders to the police, Operation Safehaven Nigerian Army, DSS, Civil Defense, NSCDC and other collaborating agencies to reinforce and contain the situation without further delay. According to Sieg P. Egbuka, the meeting basically reviewed the fallouts of the events within the last 48 hours where there were attacks specifically at Jebo Miango. From the records at Saturday night, the 3rd of July, 2021, five people were killed while about 85 buildings were burnt. The next morning, the attacks still persisted. From intelligence, the attackers and militia brought out of the state. So you get notified about fresh news updates.