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The for materal pue 1 costs can mouoq cconomical onosd top the the for L bottom The the 1 Ji can L VL Your...

Question

The for materal pue 1 costs can mouoq cconomical onosd top the the for L bottom The the 1 Ji can L VL Your

the for materal pue 1 costs can mouoq cconomical onosd top the the for L bottom The the 1 Ji can L VL Your



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In the figure below, on the left, $l_{1} \| l_{2} .$ What can you conclude about $l_{1}$ and $l_{3} ?$

We're given this magic A We're universe first. Me from the convertible meeting room. Without it, the determinant they You could have zero in a but not in veritable works. Check it. A convertible. What do you say? Actually, I read it down here. Let's find a determining a check of the convertible or not. 111 First, I'm gonna high road to buy world one by negative one and had it wrote to So I guess one might one minus one zero. Making one plus 21 No one here. Next I'm gonna multiply growth three by negative one times wrote to I get 111 negative. Q one is negative one. You know, the determining the mortification old the numbers in the pivot in the diagonal interment is clearly not so. Therefore, we can find a neighbor nullifying chambers through the over inside the right chambers on this side. Very eight in the side. You're a here you have the identity matrix for three by three One here is you here alone? Now we're gonna really do until this side here. It looks like this. Once we do that, we will get a members on this side Look for reduced. Well, we already thought before we're finding the determinant. Do it again. First they can about this here. 11 now weaken Can't hold this position here. Negative one. They won negative times. Negative. 101 Negative. 101 Here one. Now weaken. We can scale the throw here. We can divide the group by negative one. We get negative here. Also here. Positive here. Now I can scale road three by minus a few. Added row to cancel this The negative too. Times road Here. That positive you minus one. That's one. Make a few plus one minus one. Two zeros too. You get one Next. We just need get rid of this. We can. He gave the period. Rowing added to the first would be a bit of this one. Here. You never get here from zero and in one one to negative one plus 01 and 101 Now you get a second road out of the first road in negative. One plus two. Just one. You have one plus minus +10 You have minus to plus one. You hear? This is the identity matrix implies on this side. He had a members say in verse, should be one bureau minus one one, minus 12 and minus 11 minus 11

For six point to number thirty Li na Ni to minimize the cost from example for So I'm going to make a little note that we're looking at example for to get some of these numbers and we need to minimize the cost. What we know from example for is that the surface area is equal to two high R squared plus two thousand Taser. And and we also know that the height of the can is equal to one thousand divided by pi r squared. And here's what we know as faras the cross from the question we know that we are looking at three cents three cents for aluminum and we're also looking at an additional one cent for the height because of the scene. So what that means is we have for s the cost. Let's do so. S cost is going to be equal to the point o three, cause it's point oh, three cents. So three cents is point oh, three times two pi r squared plus two thousand are because it's three cents for all the surface area that we need and the height crossed is going to be that point no one times the height of the scene that we're paying a little bit extra for this is equal to point Oh, six pi r squared plus sixty over our And this height for this, when we distribute is going to be ten over pi r squared. So now we have the cost for the surface area and also the cost also the cost for that scene. And now to find the total Cross, we're going to add both of these a question equations together. So that means that the total Cross, which is see Total Cross, which we're going to call C of our is he hold to the cross for the surface area, which was point oh six pi r squared plus sixty. And I'm gonna write it as art to the minus one. Good to go ahead and bring up the exponents. Plus, now this is the height, and I'm gonna write it as ten over pie and instead of over r squared, I'm going to write it as art of the negative too. The reason I'm doing this is because the next step to minimize something, you need to take the derivative seventy four zero. So now we're taking this derivative. So we're gonna have point one two. Hi. Are minus sixty Are to the negative, too. Because we're bringing down the negative one. We're going to bring down the negative too. To get negative. Twenty over pi R to the night of three now twenty point twelve pi R minus sixty over R squared minus twenty over pi R cubed. We're going to bring those exponents back down to the bottom. We know that our is the radius, and it has to be greater than zero crazy. We can keep that in mind. And now we have the derivative that we need to look at. Now we're going to go toe are graphing calculator. So we're going to graph that c prime of Arc. You're going to graph it, and then you're gonna find the zero, which is the X intercept, but and you're going to try and do that using your graphing calculator. Okay, When you do that, you see that See, prime of our. So it's equal to zero when that our is equal to five point for five four. So now you find the radius that will minimize your car

This video's gonna go through the answer to question number 11 from chapter 9.3. So ask to use real reduction to find the inverse off the matrix. That 11 one 121 Thio three. So So we conform the combination matrix with the identity and they tried refugees. Okay, so if we subtract to you off the top equation from the bomb equation, then we're gonna get zero one that to you, minus 20 Maybe it's gonna be minus 201 on the inside. And if we should bottle subtract one of the first question from the middle equation, that's gonna be zero That's gonna be one on that's going to zero months. Well, on zero on me, the top equation as it is, Savior zero. Okay, so now we get to be a stick in court because on left inside the bomb equation on the middle or after the bottom row of the majors in the middle of the matrix. All the same, which means that the ah, the row is off the matrix linearly dependence, which by their a born in the book, means that er the identity that's all right with me

But we're continuing on with definitions here. So for this one, they want us to know what the lowest point on this graph is called. So we call this the minimum point, but it actually has a specific name for quadratic functions in a quadratic function. This wait where we change directions, otherwise known as the minimum of the maximum, has a very special name on that special name is the Vertex is the point where I change directions. It's always minimum or maximum value, and it's also kind of the point in the middle of the graph where I mirror itself. So you'll notice that the left into the right there the same kind of graph. But that point in the middle of the Vertex is kind of the middle of the graph, so that point is always referred to as the Vertex. Whatever the middle point of your poor Abdullah or quadratic function is


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