So we have some, uh, Karina is going to be taking some type of an exam, and the exam has short answer, and it has essay questions, and they give us some information about this. They tell us that, let's see, they tell us that the number of points you get for a short answer is you get 10 points for a correct short answer and 15 points for a correct answer with an essay. And it says that the amount of time it takes for each one, the short answer will take three minutes. She approximates, and essay is going to take twice as long six minutes. And we know that the total amount of time. So the total time for this exam is to finish this 60 minutes, so you can't use more than that. And we know that the number of questions. So let's say I let X stand for this is the number of questions of each type and let's say you get a T on there. And so let's say X is the number of questions we have that are short answer and why is the number of questions we have that are essay and we know that we have to have less than or equal to 16. So 16 questions is the most we can have. So we have quite a few things that we know. We know that X has to be greater than or equal to zero. You can't have a negative number of questions. We know why has to be greater than or equal to zero. So our graph would end up being in the first Quadrant. We know that if we take our number of questions x plus, why that we can't have more than 16 of those questions. And we know that if we take three times X, that's how the amount of time we'd spent on our short answer questions and then take our six times why That's the amount of time we spend on the essay questions that that can't be more than 60 minutes. So here's our little inequality system that we have to deal with, and we know what do we want to maximize? We want to maximize our points, and we're supposed to assume that when she answers the question, she's supposed to get him right, So this is gonna be points. I'll just say the points in terms of X and Y. We know that will take 10 times the number of questions she uses that are short answer and 15 times the number of questions that are essay. And this was what will maximize. So we'll come back to this after we draw our love graph. So let's draw a graph. And so we know we're in the first quadrant and we could grab some with intercepts. And if I again if I go along, Uh, let's say this is 2468 10, 12, 14, 16, 18, 20 and 2468 10 Again, I'm going to do some things. Algebraic Lee. 12, 14, 16, 18, 20 All right. And we know that if we grab were in the first Quadrant, so I won't. I won't shade those areas, but we know our only first quadrant here. Let's grab by intercepts the linear, the linear pattern. We know if we let this y b zero X will be 16, so we would have Let's grab that first one in red. That's 12, 14, 16 and same thing will be true if we let X be zero. Why would be 16. So 10, 12, 14, 16. And now let's see if I can actually draw this kind of a straight line, okay? And that would be shaded downward. Now the other relationship again. If I go by intercepts, if we let y be zero that and find what would make three times what would give me 60 that's gonna be 20. So let's do that in blue. So if why is zero X would be 20? So there would be an intercept right here. And if we let exp zero, then why would have to be 10 and again I'm going to try to draw that line, and it intersects. And sorry that didn't go so straight. But that's okay. That's okay. You could do it on paper and use a ruler I'm doing on a digitize er and can't get that that well, we need to know that point now. I can't tell for my graft because I haven't drafted carefully. But when do X plus y equals 16 and three X plus six y equals 60. Where do the Intersect? Well, I'm gonna take that equation and just gonna multiply. Divide everything by three. So that becomes X lost two y equals again, dividing by three dividing by three dividing by three and I get 20. And then I am going to subtract thes equations. And when I do, the X drops out two y minus Y gives me why and 20 minus 16 gives me four. So there's the y Coordinate. This should be at four, which actually looks pretty darn close. That coordinate there is four. And then we need to figure out what the X coordinate is. And if we go back to this equation, if I plug for in here, we find out X would have to be 12. So for this point, X would be 12. So this coordinate is the 120.12 or now that is again, less than or equal to. So this blue area would be shaded down like this, and our intersection zone is in here. So this is one of the points that's possible. Zero can. 00 is definitely not. You don't wanna have, uh, no points at all, But that's a part of the intersection of that little, uh, quadrilateral that we've formed. The 0.12 4 it might be is gonna be a point. We're going to check and then this point, another endpoint 16 and zero. So let's see where we get the most points and we're gonna go back to this equation. So the points for this equation lips and move just at the same time. How about I do it in green? The points that we get here are going to end up being 10 times 10 times zero, which is going to be zero plus 15 times 10 is gonna be 150 we would score 100 she would score 150 points if she had no short answer and just go for all essay. Now let's see what happens with this one. We would have 12 times 10. There'd be 120 points from short answer, and there'd be four times 15, which would be 60. And if she does 12 short answer in four essays she would have if she got them all correct. Should have 180 points. And if she does this and says, you know, forget those essays, they take me twice as long. I'm just going to go short answer while 12 times 10 is 160 so the best is right here. She can maximize her score by answering 12 short answer and than four essays. And if she gets those questions correctly calculated, she would end up scoring 180 points, So that's the best thing she could dio.