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what values of r and w minimize the value of 3r^2 + w^2 subject to the constraint 6r+13w=78?

This video, we're gonna go through the answer to question number 27 jumped 2.6 so as to sell off this differential equation as a binary differential equation. Let's write it in binary form. You got the our defeat, sir. Matters to a theater, uh, is equal to our squared. Overthink this word. So you see, N is equal to too, So we should. Substitution V is equal to ask minus one. Therefore, the R Beefeater is equal to minus one over B squared Devi Get better. Is that four minus one of the squares DVD Peter minus two over theater times. One of the key is equal to one over the two squared That's one of the squad. Many times the whole equation by minus V squared when we get to be a Beefeater plus two over the to be is equal to one over you two squared minus what it's good against. The integrating factor for this is the beater bar of the integral of two over theater, which is east the power of two natural Amphitheatre, which is the two squared. So therefore Hey, Peter of C two squared D is equal Thio minus one over Peter's square times, these air squared just minus one. Therefore, theater squared V is equal to minus. Peter could see their forces dream back. For our that's just gonna be equal. Thio Feet squared over C minus Tita. And those are our solutions for his defense equation.

Well this probably would like to minimize four R squared whilst W squared subject to for our plus 13 Ws 52 mm. Now, in order to maximize or minimize something, you need to get it in terms of one variable and Dylan calculon, it's always just one variable here. Here we have to Yeah. And then I get this in terms of just one variable. Let's solve this for our so let's take that 13 W. Over. So four are is 50 to minus 13 W. So are is 50 to minus 13 W Over four. Now let's put that in over here for our so we have four times 50 to minus 13 W over four squared plus W squared. And so now in green here is what we're wanting to maximize. Those are minimized. Now, in order to minimize this, you take the derivative instead of equal to zero. But before we do that, let's clean this up a little bit. And so let's expand 50 to minus 13 W. Divided by four square. Doing so, remembers that florida Outside systems four times 169 W squared over 16 minus 169 W Over two plus 169. Again, we just foiled out what we had there and princes plus W squared, distribute that for him. And that'll give us 169 W. Squared over four minus 169 times two. Because two of the tools were cancelled, uses 3 38 W. And then 1 69 times 46 76 plus W. Squared. And so this is what we're wanting to minimize. Now, we could combine the W squares here. Let's just leave it like it is and go and take our derivative. So this is what we're wanting to Mac to minimize. And so take the derivative and so are derivative. Here will be 1 69. W over two minus 3. 38 out of 6 76. 0 plus two. W. And then we set this equal to zero. That's all this. If you multiply everything by two to get rid of that fraction. Be 100 and 69. W minus 6 76. Lost four W zero. It's 173. W 6 76. So W was 6 76. Right over 100 73 which does not reduce. So this is the value of W that would minimize. Now from above, we know our is 50 to minus 13. W over four. And so are just 50 to minus 13 times 6 76/1 73. And this is a lovely number over four. So if you evaluate what you have on the right hand side, this will give us 52 over 173. And so the value of w that would minimize it at 6 76/1 73 on the day of our is 52 of once every three.

For this problem of our goes one plus Jose data. We are first going to graph it. Let's grab it right over here and we know this is actually a polar graph. You know, it's polar. From the fact of in fact it's trick and from what they did in the problem. So I'm gonna call this my center. Yeah, if you graph it on any graphing device that you have, it's actually going to turn out to something like this. It has a little pinch right here. This is something that I call a cardioid. Um and a cardioid is one of the few polar types of graphs that are out there other than say circle. And we know a cardioid when we see it's almost a circle. But not because of this pinch, a cardioid, it's met when it's typically in the form of this type of equation given it has to be in the form of one plus or minus coat Cynthia to or one and plus or minus same data, it cannot be 10 because if we did tan then this would be spiraling all over the place. But And you see that A I put right here now the A. And this problem is just one. There's always going to be a one in front of a trick if nothing is there. But in case there is another number, there a has to be greater than zero to like I said in this problem a close one. So it's always important for these to meet the requirements of a cardioid. Now, in order to go on about so much uh putting it into its summer trees and putting it into its Cartesian form. What I'm gonna do is draw out another craft. This time it's actually going to be something that resembles a polar to a Cartesian. So right here, I'm gonna do is draw disappear here. So this right here is actually are this is X. And this is why, as we can imagine with every triangle, you have a data. So this right here is what I call Polar to Cartesian. So when you were doing Polar to Cartesian or even sometimes to a Cartesian to Polar. Typically not though it's we're always referencing to this graph. When we're doing a Polar to Cartesian, we want to make sure that we're doing it in the right fashion and sometimes they might need a visual representation to make sure we're doing it right, especially without these trig equations in front of us. So that's why I dropped this graph is because oh, it might be easier to reference, say a triangle instead of all these other equations in front of us. That can get us quite puzzled at times. So what I'm going to do next is actually rewrite our equation. But this time I'm going to right out some other key equations off to the side. So we have art squared, let's X squared plus y squared. I know that we also have tan data, bulls, Y over X. And then we know that X equals our coastline data. Why equals r signed data. We're working with points. It's always the co sign fine. So those are typically some equations or something. So you want to have an art toolkit, whether it be, like I said polar to Cartesian or Cartesian to polar. So one sweater going to transition this to Cartesian. One of the first equations. Well, what we want to do is take probably this one, this one and this one, it's important to distinguish which ones you want. And the reason why is because you don't want you just get yourself too much in a struggle. Because we know we're not working with the point, we're not working with tangent so why even look at them or whatever. I'm only going cross them out so you know they exist. But we want to make sure we distinguish which ones we have. So with this in mind I'm going to actually rewrite this equation right now because sometimes we don't actually use it in its exact form. What we do is we actually square rooted. And the reason why we do is actually to just get it to our okay, we don't want to have to deal with R squared if we're trying to find our and so what we're going to be doing first is with this equation. Our first step. It's too multiply both sides are I'm gonna be abbreviating the steps so we'll have R squared equals our times one plus co sign peter. If we so break this out then we'll have our plus our coastline. The art scored still stays the same. No we got a substitute. That's our next step two. And we want to get rid of these ours. Especially because whenever you see A. R. It's mainly in polar form. So what we'll be doing as far as they are squared, we know that. Okay we have the expert. That's why I squared the or in itself. That's why we have the square root form X. Squared plus Y. Squared plus our cosign theta. That's an X. And then that's actually our final problem. So it may seem so many excuses lives. It's like oh my gosh there's so many squares believe it or not. As long as it's an X. And Y. Form, that's our answer. So knowing of everything there, you could try to re put it back into Polar to verify your answer. And sometimes teachers do encourage that to double check to put it back into polar form. And then if you're starting a petition, Parton Polar put it back into Cartesian, that's up to you on your own time and everything like that. So I do want to encourage you to do that on your own time and that's actually how you solve a polar equation into Cartesian and graf eh polar equation as well.

Here, we want to minimize three R squared most W squared subject. The six are plus 13 W. The 78. The first thing that we were wanting to do here is just an optimization problem is solved for one of our variables here on the right and it doesn't really matter which one that you solve for. I'm gonna sell for our so six are the 78 minus 13 W. So our 78 minus 13 W over six. Now put that in over here, this this is three times 78 minus 13. W over six squared plus W square until we just substituted that in here on our left. Now, before we actually do any calculus of this, let's simplify this one more. And so let's expand 78 minus 13 W over six square. It's also be three times 1 69 W squared over 36 minus 1, 69. W over three plus 1 69 and just foiled out what we had inside the parentheses there. I just went and squared. I still have a plus that other w square distributing the three. And this would be 169 W squared over 12 minus 169 W. Lost 1 69 times three. It is 507 plus W square is what I have in blue here is what we're wanting to minimize. That's what we're wanting to minimize. So to minimize it, you take the derivative in saturday 40 So the derivative of 1 69 W squared over 12. There's 1 69 W over six. The rate of 169 Ws 1 69 through the 5070 And derivative of W. Spertus to W. That's why we set that equal to zero now, 1 69/6 plus two gives us 100 and 81 W over six so that it holds 169 divide by 1 81/6 on the inside. Let's give you a really ugly number here. But this is going to tell us that W 169 divided by 1 81/6. And that gives us this wonderful number of 1000 and 14/1 81. And so that's the value of the W. That minimizes this. They also want the value of our and so we need to play this back in. Well, our is 78 minus 13. W over six. So are is 78 minus 13 times 10 14/81 over six. And we just pulled down the value that we knew for W. So if you type this in on a calculator and that's what I would recommend doing. I would not recommend trying to do that in my hand. It's going to be a real pain. This will give us ours 1 56/1 81. So here the values of our nW that minimize that original expression.


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