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Question 110/4 ptsDetailsMinimize2 = 5r + y 3y + 4 24 4y + 32 2 24 Subject to y + I 2 2Minimum is...

Question

Question 110/4 ptsDetailsMinimize2 = 5r + y 3y + 4 24 4y + 32 2 24 Subject to y + I 2 2Minimum is

Question 11 0/4 pts Details Minimize 2 = 5r + y 3y + 4 24 4y + 32 2 24 Subject to y + I 2 2 Minimum is



Answers

The minimum value of $2 \mathrm{x}+3 \mathrm{y}$ subjected to the conditions $\mathrm{x}+4 \mathrm{y} \geq 8,4 \mathrm{x}+\mathrm{y} \geq 12, \mathrm{x} \geq 0$ and $\mathrm{y} \geq 0$ is (1) $\frac{28}{3}$ (2) 16 (3) $\frac{25}{3}$ (4) 10

Okay. This question gives us this objective function to optimize subject to these linear constraints. So plotting this region and Daz Mose, we come up with this graph and since we have less than signs everywhere, that means that the region we're worried about is this part right here in the first quadrant. So what this means is that our vergis ese are here, Here, here, in here. And this defines a regent. So this vertex rate here iss 50 This vertex right here is the origin. This Vertex right here is 36 and this Vertex is 0 10 So now that we found our four Vergis ese, we just need to test them into our objective function. So Z of 00 is just zero plus zero. Which zero then see of 0 10 is equal to zero plus 10 which is equal to 10. Then Z of 50 is equal to two times five plus zero, which is also equal to 10. Then Z of 36 is equal to two times three plus six, which is equal to 12. So what we can see here is that our minimum value is zero and our maximum value his 12th. So now that we have this information, we can just fill in that our maximum value is 12 at the Vertex 36 and our minimum zero at the origin. And just to note here we see that 0 10 and 50 for deceives actually have the same value. So that means that anything on this lavender colored line in our feasible region, we'll have the value of 10 also, but it doesn't really matter. Since it's not a maximum in, it's just something worth noting.

In order to find them in and max size of the subjective function sketched is the constraint graph in order to find the point so need to pluck it. So if we took the first inequality and we find excellent intercept, we can plug in X equals zero and that will give us zero plus. Why is less honorable of 15? And that's at zero. Calling up 15. So that's right here. If we plug in Y equals zero, we'll get three. X Plus. Zero is less than the wrinkles of 15. So if we divide by three less than or equal to access that we got last X is less than marigold of five. So that gives us the point. Five. Comment. Zero dots right here so we can connect those two points to get that line. I'm looking at the next inequality. If we plug in X equals zero will get zero plus three wives a Syrian golden 30 if we divide both sides by three. But why is this during Golden 10 which gives us a will? Calm a 10. She's right here. If we plug in why? Well zero. Then we'll get four x closed zero so cynical the 30. So if we divide by four who got X is less son Erin. Cool too. 30 over four, which simplifies to 15 over to if we divide both by two. And that simplifies to 7.5. So back is this the 0.7 point five comma zero, which is right. Hear about so we can connect those two points. So now when you figure out which area is shaded so we know that accent where I must be greater than you're able to see her out, we know that excels. Listen, they're equal to five and less on Miracle to 7.5. So x is gonna be on this left side of this line somewhere. Can you know that? Why is less than gold of 15 in listener Gold attendant? So that means we've got a shade in this area right here. So, in order to friend the minute X values, we need a plug Indies bordering points. But we don't know this point right here. So to find that cancan set up a system of linear equations, um, that could make it easier if we can cancel out on X or y so Okay, so let's take one of these inequalities. And if we look at these Buffy's inequalities in order to cancel out why, for example, can multiply this top one by three Everything right there. So if we subtract it and well, 53 you get three times three is nine. That gets us nine x three times Honest three makes us three Why? And then three times 15 is 45. OK, so freeze trucks that would get four x minus nine X is negative. Five x We got three. Why Ministry y zero and then less than an apple to 30 minus 40. Private's name that 15. So if we divide our five, we will get free. So that puts us Act three for acts and then from the Why Val, you can plug it back in. And so one of these inequalities So C four times three. It's 12 plus three. Why is listen I equal to 30? So 12 plus green? Why it's less than uncle that 30. If we subtract 12 we get three. Why is less than an equal to a team? And then if we divide by three, get why is less than or equal to six so that I could disappoint. Three comma six. And that's this one right here. Sonata from them in an exercise we can plug in these four points into this objective function. So let's start with to Euro crowd Zero from that and I can't see equals zero. Okay, now let's try zero comma. 10. If you plug that in God see, he cools zero plus 10. Now, let's try, uh, five. Come on. Zero. If we heard that in, we'll get C equals five times five is 25 plus serum. And then our last 0.3 comma six got in and we get C equals three from Privates 15 plus six, which puts us at 21. So it looks like her Max is C equals 25 at the point. Five comma zero in our men is his equal zero at the 00.0 common zero

Okay for number nine, we're finding the minimum thes two equations exist simultaneously. I mean, we both true, so we can combine them to make one, and that's going to show us our Q. Okay, so we know that X plus Y equals by weaken. Reduce this toe one variable, and then we'll be able to take the drift of it easily. Um, one way to think about this is that why was equal to five minus X, right? So we can go ahead and put that in up here and rewrite the Q equation like this and then, uh, plus three times this whole business squared, which will give us something that is Onley in terms of X, which is exactly what we want. But we got to simplify it a bit here, too. Two x squared, and it's good. Distribute what? Step by step. Phillips. Wait a minute. There is a squared right there. We can't do that. You have to square it first. Eso do three times, whatever the square of that is, which would be 25. It's the first inside plus the outside to be, um negative five x negative five x on the last would be plus X squared parentheses, which is kind of going off the edge here. We can simplify that. Don't make it a lot better. Um, but first, we should probably do our distribution that I was so itching to do originally. So we have plus 75 minus 15. 0, I'm sorry. Actually, this is, uh, Samos Negative, 10 x. So we're minus 30 X. Right? And that brings us down here. You can combine her X squares to be five x squared, minus 30 X plus 75. Okay, so at this point, we can go ahead, and because we're looking for a minimum, we know that this is a situation where we have some sort of, you know, parabola. And we're looking for that minimum amount. We don't know where this is going to be on the X and y access, but we do know that the, uh, slope at that 0.0.0 so we can set Q 20 I was sorry, Q prime to zero and then that. Allow us to find it. Let's go and find Q prime. That would be 10 X minus 30. Okay, so now when we set that equal to zero. We end up with this. X equals three. Okay, that's one of the numbers. You have to go back up here and substituted to find the other one. Eso we have this one. Why equals five minus three? That would be too. So y equals two. So are ordered set Would be for the final answer. I'm sorry. Are unaltered. Said it doesn't matter. What Order these, Aaron. Three until excellent. I speak circle. How about thanks?

Okay. This question gives us a function to optimize in this constrained region here. So the first thing we need to d'oh is graph to find over Texas. So plugging into Daz Mose, we can see that our region ends up being this area in the first quadrant right here. So now we can look for where the Vergis ese are in this region. So we see that there are four of them. One corner is here. One corner is here. One corner is here and another is right there. So no seeing where these intersect. One Vertex. Is that the origin? One Vertex. Is that the point 50? Another Vertex is that the point 36 and another Vertex is at the 0.0 10. So now that we have our four Vergis is we need to plug each of these into our ejector function and find our maximize er's and minimize er's. So see of our first vertex 0 10 is equal to zero plus 10 or turn then Z of 00 is just zero plus zero and that's just equal zero. Then see of 50 is equal to zero plus five, which is equal to five, then our final vertex Z of 63 Sorry. This should be Flip Z of 36 equals three plus six, which is nine. So we can see that our minimum is zero in our maximum is nine. So our maximum is nine at the point 36 in our minimum is zero at the origin and that's our final answer.


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