So for this question, an animal shelter makes its two brands of dog food. So we have brand X and brand. Why, right? X m burn wife on brand X cost $25 for back and contains Come back UN contains So units of nutritional elements a two units of regional elements be so he has so units off a yes. Two units off elements Be yes, two units of elements C right. And now brand. Why cost $20 per bag right on it contains one unit off elements A contains nine units off elements be on three units over elements C now the minimum required amount for no trace A, B and C ah, 12 units. So for eight, the minimum required amount is 12 units for be the minimum requirement. But for me is 36 Units I foresee is 24 years now. What is the optimal number of bags of each brands? That should be mixed on what is the optimal cost. So we need to find the optimal way for making the two brands X and y as food so that the minimum requirements for nutrition is which is this the minimum required Mexes met on. Also, the cost is optimal, so that's what we're trying to do. So let X speak. Number of runs off X on. Why building number bags off? Why? So let's more eggs. Be number of bags of rent X now Why be box of round? Why now? Therefore, the function is going to be C is a quote to 25. Close 20 right? I'm sorry, 25 x was 20 way, and that's because it's 25 for you need off eggs and 20 units $2020 per units for why? So that's that we have to find a minimal cost now on the three constraints can be converted into linear inequalities as two experts. Why is Quintana equal to 12? Because if you go back, we can see that we want to. Units of a four x one is of a for why, and then we want the minimum is 12. So it has to be greater and swelled. So I'm gonna follow that same structure and creates three linear inequalities. So it's going to be the next one is going to be two X was nine way greater than equals to 36 the last one is gonna be two x was three y. Lieutenant equals to 24. Now, this this is a number of bucks used kind of a b negative. So we cannot get more constraints of eggs with 10 and zero and white with another, because with an unequal necessary, because the number of bucks can never be negative. Right? So Well, we know my good to find the Adriatic timing by the constraints on. Then we can see that if you draw the area to tell me by area determined by the constraints. So these are the constituents. Where am I gonna find area as determined by the class drinks? So the time you should look like this. Well, why Over here? Yeah. So, yeah, the first point will be zero. Come on. Four, then. Zero comma. Eighth zero comma 12. They were gonna have a point here on four on four coma for on the point on three. Comma six. This is a Then this points joins to this away. Then this point joins all the way to this one too. So that should look like this. And then we have six trauma zero. Then we have 12 from a zero. No, we have 18 comma. Zero. So we kind of all these points? No. The point on Ni Komatsu and this is point B. So I'm going to connect the point. This is to expose nine. Y is equal to 36. This point is two x close three y Is it called 24 and this is two eggs. What's wise invoices? So this is what this is the area determined by the classrooms. And this is what it looks like on a graph. So they are moving on now. The figure. All right. We can tell that the point is the point Where lines one on three Inter set. If you go back to the table to the they had questions, inequalities, dishes. It is a plane with on one and 3 to 6. So because this is a question three and this is a question one. So if you go back there, we can see that that's when they eight or six a on from equation one two X is equal to to actually cause the 12 minus. Why? So make this work patients to sorry equation for Because if you go up to the graph. This is the question one, this integration suit in question. Three. Rhetta on this situation to. So this point is where the intersect this to the Intersect, So that's official. One. So from equation 12 x right is equal to 12 minus y, and that's Equation four. Now from Equation three two. X equals toward 24 minus three y And that's equation. Would five. Now, at this subsection points X values at the same so we can equate, you know, equals show four on five together so we can get, um, 12 minus. Why is it close to 24? Minus three way then, even though bring it over to the other side, it will be It will be, you know, 12 minus 24 is equal to minus three y plus y and then minus 24 minus 20 No minus 12 and the courts and minus two y and they were divided. Why is he quotes? What six. So you're going to substitute? Why is the course of six in a Christian for so what you thought it serves us See question for we get two X equals 12 minus six x the quote six or what? I'm sorry. Extra cost of three because we divide six. Fight about two. So So now we know what X is, you know, a wise. So the coordinates off the intersection points A is worth three commerce sticks. Now the points be in that is this protection off two or three. You go there. Business and sectional words. So three. So now we can see that from equation too. Confusion, too. We can see that two x is equal to 3. 36 minus nine y And this equation six from aggression three two eggs because 24 minus three wide. That's the question seven. Now intercepting the points. Um, since X values are the same words because, see, six because is seven. So that's six minutes. Nine wise, of course, it's 24 minus three white. Right, So we're going to get that's six minus 24 is, of course, 93 wide close nine wife and then that's it for 24 is 12 on, then its request to six way. And why is it close to two? So because you know why is supposed to subsidies value any question six, which is that so from there to exit goes to take six minus nine times two. I meant to actually supposed to get the six minus spacing and then X is equal to exit. Wants to actually wants to 18. And it X is, of course, tonight so that the body of knife So the coordinates off the point that's a short points off a is my comments to so moving on. We found the coordinates right on at 5. 36 of the region formed by the constraints. The function who have values as following. So at zero comma, 12 Sesay courses 25 open bracket zero was 20 couple of records. Why is what's wolf on? This is going to give it support. See at the 0.3 Commerce six I 0.3 from my sixties it close to 25. Open back at three lost 20 open records. Six on this is 1 95. So this is the minimum while you off See And then my at my Komatsu C is because the 25 open work in nine was 20 open brackets to on this is to 65. I have 18 comma zero Sesay close to 25 open brackets. Kate's name close 20. Open bracket zero on this is the course of 4. 15. So if you look at see what this really well volumes see, and this is the maximum value of C zero. So if we can conclude that the minimum cost, of course, when you know there are three bucks of eggs on six bucks of white, a bolt element Marcoses wanted to five and then the maximum cost, which is this off course. When they're 18 bugs off, it's in box off. It's Andrew Bugs. And what? But you want to find the optimal cost. So the optimal cost cost is nice. 1 95 on it, of course, with three buds off s and six bucks and white. So Dr Marcos is 1 95. The news. Your clothes roots. Three bucks. Oh, it's wild on six bugs off, wife. That's the answer.