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CCadbury wants to change their main ingredients with A and B_ TThere are four kinds of the ingredients per unit that has A and B:IngredientsD1 D2 D3 D4ICost/kg12 21...

Question

CCadbury wants to change their main ingredients with A and B_ TThere are four kinds of the ingredients per unit that has A and B:IngredientsD1 D2 D3 D4ICost/kg12 213Cadbury needs units of A and 2 units of B every production batch: Which type of ingredients and what amount should be purchased to provide the cheapest ingredients?

CCadbury wants to change their main ingredients with A and B_ TThere are four kinds of the ingredients per unit that has A and B: Ingredients D1 D2 D3 D4 ICost/kg 12 2 13 Cadbury needs units of A and 2 units of B every production batch: Which type of ingredients and what amount should be purchased to provide the cheapest ingredients?



Answers

Maimizing cost Two substances, $S$ and $T$, each contain two types of ingredients, I and G. One pound of S contains 2 ounces of 1 and 4 ounces of $G .$ One pound of $T$ contains 2 ounces of 1 and 6 ounces of G. A manufacturer plans to combine quantities of the two substances to obtain a mixture that contains at least 9 ounces of $\mathrm{I}$ and 20 ounces of $\mathrm{G}$. If the cost of $S$ is $$ 3$ per pound and the cost of $T$ is $\$ 4$ per pound, how much of each substance should be used to keep the cost to a minimum?

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.

All right. This question gives us a lot of information, so let's just write it down as we read through. So what we're gonna need is an objective function, which is our price. So it tells us that our price they were spending on these bags is $25 per x, plus $20 per y. Then we want our constraints on the nutrients. So, nutrient, eh? We get two units for every X plus one unit for every why, and that has to be bigger and 12. Then for nutrients, be each bag of ex gets too, and each bag off. Why gets nine in the minimum for that has to be 36 and then for C bag Ex has to in bag. Why has three and that's 24 needed, and some or constraints is that you can't have negative bags of dog food. So we're trying to maximize Sorry, minimize the price spent because we're looking for greater than for all these colored lines. So that means that are feasible. Region is anything in here, so it's unbounded, right? Because you could just buy 1000 bags of each food and be set. But for a minimize. Er, what we're looking for is thes vergis, ese. So there's only two options that really work for us in these points are 92 and 36 So let's test the price at each of these and see which one is better. So see of 36 is equal to 25 times three plus 20 times six and punching that into our calculator. We get 1 95 then to test our other point. See of 92 is equal to 25 times nine plus 20 times two, which gives us to 65. So significant difference. So this one is the cheaper option. So minimum price is equal to 1 95 and you want X to be equal to three and why to be equal to six. And this is the solution to our optimization problem.

Hello. We're working on a linear programming problem here. Um, a real life one where we have a cookie factory that wants it. So two different kinds of cookies and packages of 6 to 12 and so there needs to be at least three types of cookie in a package, and so and were given some other information as well. So I want to make a table here. Um, we have chocolate chip cookies and we have peanut butter cookies. So these are the two types of cookies, and then we have some information about these. So to make a chocolate chip cookie, it costs 19 cents. And to make a peanut butter cookie, it's gonna cost us 13 cents each. And then when we sell them, a chocolate chip cookie is gonna sell for a cost of 44 cents each and a peanut butter cookies gonna sell for 39 cents each. Okay, so that gets started. Now, the other thing we need to know is the fact that, um, what we're trying to do is figure out what's the maximum profit? How many of each type of cookie do we need to sell the maximize their profits So we're gonna make chocolate chip cookies. They're gonna be our X coordinate peanut butter cookies. They're gonna br Why coordinate for everything that we're going to do here? Okay. And so the first thing we need to do after we have set up our, um what are variables are going to be? We need to write a system of inequalities here that represents the situation. So, toe, help us do that. The first thing. And since we know that X is gonna represent the number of truck chip cookies and why is the number of peanut butter cookies? I know that since I have to have at least three in every package, I know the X and Y both have to be greater than your equal to three because we have to have at least three in each package. We also know that when we add these two together, it has to be either greater than or equal to six cookies because ace element packages from 6 to 12. Or it's gotta be when you add them together left and her equal to 12. And so this is going to represent our ah system of linear inequalities here, where we know there has to be three cookies of each, at least in each package. So we gotta set those restraints, and then we have to have at least six cookies, but no more than 12 okay? And we want to find what is the best combination of cookies to maximize their profits. Now, we could talk about profit here real quick. Profit when you sell these things is gonna be based on, um, your selling price. Subtracting your how much you make right. So if we take 44 subtract 19 ah, 44 subtract 19. So you're making 25 cents on every chocolate chip cookie and 39 minus 13. You're making 26 cents on every peanut butter cookie. And so our prophet equation is going to be, um, f of X Y. Our prophet equation will be, well, you're gonna make 25 cents for every chocolate chip, plus 26 cents for every peanut butter. And so now what we're gonna do we have our we have our linear assist inequalities, letting your system of inequality. We have our prophet equation. Now we're gonna graft these inequalities and figure out where feasible region is, and then figure out where we can maximize our profits. So I'm gonna click over here two days most. And, as you can see, already have all of our inequalities typed in X plus Y is greater than equal to six x plus y squared equals 12 and then X is greater than three. Greater than equal to three. Insane With why And so as I used as most dot com and I start to graft, ese, we're looking for Where are these graphs all shaded together. So as I look, I went a little too fast. Heroes go a little slower. So here's our first equation. Here's our second equation. So you can see. Um oops, I messed that up. This should be less than or equal to 12. There we go. Okay. So now we're double shaded in this region right here. Um, our X is greater than equal the three that's gonna be on the right. So now we're looking at this region right here, the darker region, And then why is greater than equal the three that cuts it into this triangle right here. And we confined our feasible region with these points right here. the points 39 points. 33 in the 0.93 Okay. And so what I'm gonna do is I'm gonna take these three points, and I'm going to plug them into our profit equation and figure out if we have a maximum or minimum. I'm gonna need another whiteboard here to make this a little more clear. So if I draw this out Ah, these are gonna be in my X y coordinates that we just found. This is gonna be my profit equation that we talked about 25 X plus 26. Why? And this is gonna be our total profit, given the combinations of cookies that we found. So the three points that we found worth 39 and then we had 33 and then we had 93 And all we do is plug those numbers in 25 times three plus 26 times nine. So 25 times three, 75 plus 26 times nine. This is gonna be 309 sense. We've got 25 times three plus 26 times three. That's gonna be 75 plus 26 times three, which gives us 153 cents. And then we go 25 times nine plus 26 times three, 25 times nine plus 26 times three, and that gives us 303 cents. So when we look at this, we look at this row. This column, I should say we're looking for the highest number of the highest number is 309 cents, which means that this is our maximum profit of 309 cents are max. Profit will be found when we fell three chocolate chip cookies and nine peanut butter cookies. And that's how you go about solving that problem. Thanks for listening.

All right. So if we are, uh, buying a pack of 10 bagels for 3 69 we want to know how many 12 would cost. We considered this ratio 10/3. 69 equals 12 over X Cross. Multiplying to get 10 times x is 10 x 12 times 3 69 It's 44 28. And if we do right by town, then we're just moving the decimal point when you get a 4.4 to 8 trials to about $4.43 that would be options.


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