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Score: 8.33/15 11/15 answeredQuestion 9G0.33/1 ptScore on last try: 0.33 of pts. See Details for more_Next questionTry a similar question You can retry this questio...

Question

Score: 8.33/15 11/15 answeredQuestion 9G0.33/1 ptScore on last try: 0.33 of pts. See Details for more_Next questionTry a similar question You can retry this question belowSolve: 4/r - 2+2 > 12The answer has form: OA < x < B x<Aorx > BWhere Aand B~16;Question Help: Omessage Instructor

Score: 8.33/15 11/15 answered Question 9 G0.33/1 pt Score on last try: 0.33 of pts. See Details for more_ Next question Try a similar question You can retry this question below Solve: 4/r - 2+2 > 12 The answer has form: OA < x < B x<Aorx > B Where A and B ~16; Question Help: Omessage Instructor



Answers

$A$ professor grades an exam by awarding 5 points for each correct answer and subtracting 2 points for each incorrect answer. Points are neither added nor subtracted for answers left blank. What is the exam score of each student? A student answers 14 questions correctly and 4 incorrectly while leaving 2 questions blank.

We were asked to solve a problem using linear optimization. We're told that we're taking a test with computation problems. They're six points each and word problems that are 10 points each. We're told that we can complete a computation problem in two minutes. The word problem in four minutes, and we're told that we have 40 minutes to take a test. You can answer no more than 12 problems and, assuming all our attempted answers are going to be correct. Were asked to find how many of the chip problem that we should answer so that we maximize our score and to find the maximum score. So let's define a few variables. Let X be the number of computation problems attempted and let why be a number of work problems tempted and let ZB the score total score Well, we have that the total score physical to score from the attempted computation problems, which is going to be six times X plus the score from the word problems. She is 10 points per problem 10 times why this is a linear function of x and Y, and this is our objective function which we're trying to maximize. Dysfunction has some constraints on it. So the time it takes to complete these problems is going to be the time to complete computation problem or the computation problems, which is two minutes per problem. So two times X plus the time required to complete the word problems, which is four times why. And we're told that this total time must be less than or equal to 40 minutes. The number of problems answered this simply member of computation problems X answered, plus the number of word problems answered why? And we're told that this must be less than or equal to 12 answers. So we have these constraints on our objective function. One other constraint, which is implicit in problem statement is that X and Y are going to be greater than or equal to zero. This is because we can't answer and negative number of questions. We can't be adding questions to the test, and so because X is greater than equal, why X and Y is greater than or equal to zero follows that the solution to this system lives in the first quadrant. I grabbed the first inequality and red to do this cell graft to X plus four y equals 40. Why is it called a zero? We have that X is equal 20. So we have an X intercept at 20 zero. And if X is equal to zero wise, it was 10. So if the y intercept at zero 10 take these two points by a solid line and using the origins a test point, we do have that zero is less than a quote of 40. So everything to the left of this red line is a solution. A graph The second inequality in green. To do this so graft X plus y equals 12. We have that we have excellent intercepts at 12 00 12. Pick these two points with solid line and then using the origin is a test point. We have zero is less than or equal to 12. So everything to the left of the Green Line is a solution. I'll draw these solution to the system of inequalities in blue. We obtain it a quadrangle which has vergis ease at the origin at 12 0 at 0 10 and to find the intersection between the green and red lines. Let's solve the system of equations two x plus four y equals 40 and x plus y equals 12. So this system subtract twice second equation from the first equation. So we have four y minus two wise to y and 40 minus 24 This 16 so that why is equal to eight and back. Substitution X is equal to 12 minus eight just four. And so we have a point of intersection had for eight and just looking at the picture. This seems to be accurate from our drone. And now let's cap take the value of Z it each of these various ease at the origin. ZZ Quarter zero at 12 0 z is equal to six times 12 which is 72 at 48 Z is equal to six times four, which is 24 plus 10 times eight, which is 80 24 plus 80 which is one of four. And at 0 10 z is equal to 10 times 10 which is 100. We have by a the're, um that if a maximum for this objective function exists over this blue region, the maximum will occur at at least one of the vergis ease. And so, looking at our calculations for Z we have that the maximum over this region is 104 points, which occurs at 48 And so we have that in order to maximize our score, we should answer four computation problems. Yeah, and eight word problems. And if we do so, will attain the maximum score, which is 104 points. And this is

So I've using just point method the given an equal tease. You minus two different by scrupulously it's less than two. Now we need to have zero on one side. So let us sublime both sides off this inequality by two. So we get Q minus. Do Bye. You plus three minus two Less than two minus two. Simply find this. We get Q minus two by que plus three minus two less than zero. How? Simplifying this beget you. Minus two minus two times Que Plus three don't buy you, Leslie, which is the stands, you know, simplifying the name, but I don't get you minus two minus. Do you minus six student by you bless three less than zero on simplifying this. We get minus Q minus eight. Do it any way you bless. Three less than zero. Now let us find the zeros off the new agent as well as the denominator. So minus Q minus eight. Will zero in place. You would do minus eight on dhe you plus three equal to zero in place. You equal to minus three. Hence the zeros off the numerator and the denominator or minus eight and my industry, respectively this minus eight on my mystery. Little smart in the number line since the Raffarin function becomes undefined at quick will do minus tree. Let us mark this point, Sq, that is undefined in the number line. So the end of the line graph is shown here. So we see that it is not this. You have a ministry. Now look at this number line. They see that this two points minus eight and minus three. Device the number line 83 intervals that ISS from minus infinity to minus eight on from minus a minus three on dhe from minus 32 Now, let us check the given inequality with this angels. So consider that inequality minus Q Mei Mei ask, you might say do it by pupils three us out off you that out off. You equal to minus Q minus eight. Devoted way you less. Three. This should be this inequalities less than zero, as we have seen now, Jules, our best point from each injury. So from this interval, lettuce juice minus nine. On from the next interval, let us choose minus six. And from this interval, let us choose minus one. Now let us find the value of Are you part of you in a step once. So first, let us find out of minus nine Just minus nine. Minus or minus night minus eight. Don't apply. Minus nine plus 30. So we get mine minus eight. Do it by minus six. Thatis finest one basics. This is less than zero. Next to let us find the value of outof minus six, it's minus off minus six minus eight. Minus six plus three. Get six minus eight minus 30 minus Dubai minus three. Which is he going toe do by three. This is greater than zero. Next to the just test for off minus one minus. Soul minus one minus eight. Divided by minutes one listing. You get one minus eight. Well, by two. This minus seven. Big to again. Just an zero. Now let us test the inequality in the zeros that this minus ain't man Ministry. We shall neglect minus street because the function is not defined. Minus three. So I just find the value off our off minus eight. Naina So minus eight. Minus eight ministate Less since the new military zeal. You get this aceto Now you see that all of you must be less than zero. So for the best points minus nine, it's less than zero on four minus one. It dismissed an zero hence, the intervals included in the solution said, are minus infinity toe minus eight and minus three toe infinitive on. We see that the test. The zeroes are not included in the solution set, and the solution set in a duel. Tradition is minus infinity. Don't minus nine Union minus three to infinity.

And now we have you been You are about to take a test That contest computation problems off work. So computation problem six point and the world problem off 10 points and you can go competition problem in two minutes. So competence in problem two minutes and world problem in four minutes. And we have given the constant that is 40 minutes to take the test. So that means let we have to solve X and y question. So that means here this was with two X plus four way should be less than 40 minutes and no, you have no more than 12 world problems that is here. No, they did. A total problem is no more than well. And also X is greater. Zittel, why is greater zero? And now we're per sold these constant. And we have given that six x and then white. So you with this constant, I'm using gift to graph the Lena programming. So now see the points here This point is 0 10 And the point here is that is well and zero. And this point is here that there's four and it and when we go for the maximum ages are bigger than quite 0 10 for it and 12 0 when we go for the maximization, the four. And it gives the maximum region so really close to six x plus 10 way. So we have to choose for the 100.4 day. So this will be six multiplied with word that there's 24 this will give 80. That means the maximum score I can give here is 14 is a maximum. And when I choose four as the competition problem and full computer nation and eight word problem. So this is the required here.

Okay, so this problem give us that. It was a revised score. 75. It wants us to find the original score using the formula R equals 10 times the square root of the original score. So dividing both sides about 10. I'll have 7.5 equals the square root of the original score. Okay. Never mind stories. My radicals are my radical is isolated and square both sides. Okay, 7.5 squared is going to be 56.25 Think was the original score on that exam or test? Okay, thank you very much.


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