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Question 1. You are the supervisor of a math team and your employee was given the problem below. The employee provides an answer to the problem and YOur job is to d...

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Question 1. You are the supervisor of a math team and your employee was given the problem below. The employee provides an answer to the problem and YOur job is to decide if the answer is correct Or nob_ That means VOUT answer t0 this question is either Yes or No. Ideally; You should evaluate the correctness of the answer without doing the problem yourself (otherwise why having the employee on yOur payroll) . Problem: Let r1 [6,5] and r2 = [-15,18]. Put M =r,r2], B = (-48,82T and solve MT .X = BP

Question 1. You are the supervisor of a math team and your employee was given the problem below. The employee provides an answer to the problem and YOur job is to decide if the answer is correct Or nob_ That means VOUT answer t0 this question is either Yes or No. Ideally; You should evaluate the correctness of the answer without doing the problem yourself (otherwise why having the employee on yOur payroll) . Problem: Let r1 [6,5] and r2 = [-15,18]. Put M =r,r2], B = (-48,82T and solve MT .X = B Provided Answer: x =[,7" Question 2. You are the supervisor of a math team and your employee was given the problem below. The employee provides an answer to the problem and your job is to decide if the answer is correct Or" not _ That means YOur answer t0 this question is either Yes or No. Ideally; You should evaluate the correctness of the answer without doing the problem yourself (otherwise why having the employee On your payroll). Problem: Let [5,1] ad r [-4,20]. Put [ru,re]; B [[27, 16], [47.24]1, and solve MT X=B Provided Answer: X = [[4,3], [5, 1]]



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Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions (a) John has a larger mass than Barbara has. He is standing on the $x$ axis at $x_{\mathrm{J}}=+9.0 \mathrm{~m},$ while she is standing on the $x$ axis at $x_{\mathrm{B}}=+2.0 \mathrm{~m} .$ Is their centerof-mass point closer to the 9.0 -m point or the 2.0 -m point? (b) They switch positions. Is their center-of-mass point now closer to the 9.0 -m point or the 2.0 -m point? (c) In which direction, toward or away from the origin, does their center of mass move as a result of the switch? Problem John's mass is $86 \mathrm{~kg}$, and Barbara's is $55 \mathrm{~kg}$. How far and in which direction does their center of mass move as a result of the switch? Verify that your answer is consistent with your answers to the Concept Questions.

For this problem. We're looking at the ball company, which manufactures three different types of lamps, and we're hoping to maximize the profit that they're making for the manufacturing. These lamps. So three things we need to try to find from what we're told and from the chart given in this problem the first what is our objective function? Well, our objective function in this case is going to be the equation that we're trying to maximize, which is the profit. So if I pick an equation Z to show the prophet looking at my chart, I can see that my profit per unit it's $5 per each of the lamp type A that we make $4 for each of lamp B and $3 for each of lamps. See? So the coefficients in this objective function are 54 and three, and that corresponds to choice number three. Now, what about our constraints? Well, the constraints are what limits us. Um, now the limits air not shown on our chart. The charges shows us our profit and how many hours we use. But reading through the problem, we're told about our constraints. There's Onley, so many work hours per day. Department one on Lee has 400 work hours per day available in department two has 600. So those there are constraints those air what gonna limit the number of lamps that we can make. So the constraints correspond to choice number four. And our last question is, let's look at the constraints in department one. Well, if I look at that top line of my chart, that tells me the number of work hours I need in department one two hours for every lamp of type A that I make three hours for every lamp of type B and one hour for every lamp of type C. And we're told that our work hours for department one on Lee have we only have 400 a day. Remember, we saw that I'm circling that in blue. That was one of our constraints that we just talked about. So these hours in department one have to be less than or equal to 400. Yeah. Now, I used a B and C because those were the letters here. If we rewrote this with X sub one, except to an ex up three, we can see that this is a match for number number three. So those are the correct options out of these lists

This problem focuses on an apartment manager who is trying to figure out how much to rent the apartments for. He has 80 units available to be rented. So we are variables gonna be X. So we're going to let X equal the number of $20 increases to the rent. Because let's take a look at what we have. At $400 a month, every unit is rented. How many apartments? Air rented If I make X increases well, for every increase, I lose one apartment that's being rented. So one increase means I have 79. Wretched to increases in 78. 3 would be 77 so on. So this is the number of apartments that will be rented. Okay, Now, how much rent will he get? Per apartment. While he's starting with 400 and we're putting in so many increases, each increase gives him an additional $20 so one additional increase will be 422 increases is 4 43 increases is 460 so on. So this says how much the rent is per unit. We can use these two pieces of information to find the revenue function that he will get for these units because his revenue is going to be the number of units he rents out, which is a T minus x times the rent per unit, which is 400 plus 20 x. If I multiply this out, that gives me three. 32,000 plus all the outer in the inner is 1200 X minus 20 X squared. And just for simplicity sake later, I'm gonna rewrite this in descending order. So there is our revenue function. We can use this to find all sorts of scenarios. How much? How much money will I make if I rent rent out this many units? If I pep this much as my rent, what does this give me? What's my maximum rent? Lots of questions that we can answer with this to go and look at in particular First, for what number of increases will the revenue B 37,500. So I want the revenue to be 37,500 and I want to solve for X number of increases that will be required to meet this number. Now to do this, we're going to set everything equal to zero. So I'm gonna pull everything over. Um, actually, I'd like to have everything positive. So I'm going to pull everything over to the left hand side of this equation. I like to have a positive X squared term. So x 20 X squared minus 1200 X plus 5500 equals zero. Okay, We can make these numbers a little smaller. Everything here is divisible by 20. So that gives me X squared minus 60 X plus 2 75 equals zero. Fortunately, this is a fact. Herbal. Try no meal and it factors into X minus five and X minus 55. So we actually have two different scenarios. If I set these equal to zero, I either get X equals five or X equals 55. So in order to make my 37,500 as my revenue target, I either need tohave five increases of $20 or 55 increases of $20. Either one of those values will give me the revenue I'm seeking. Now, what is my maximum revenue? Well, looking at this equation back here that start equation the one that starred I put a blue star next to it. If you examine that, you can see that the highest degree term is X squared, which means this is a parabola. It has a negative coefficient, which means it's a downward facing parabola. So to find my maximum revenue, I need to find my Vertex. So let's come. I'm going to scroll up, just gonna scroll up just a little bit, do that just a little bit. That gives me a little bit more space here. I'm going to take my revenue function and it just re copying it. So I have some space toe work underneath it. And in order to find the Vertex, the easiest way to do that is to complete the square. So every term that has an accident, I'm going to factor out a negative 20 because I want that X squared term toe have a coefficient of one. So I factor out a negative 20. So I get X squared minus 60 x, leave myself some space to work, and I take my constant term and put it over on the side. It could just sit there for a little bit Now to complete the square. We need to look at the coefficient of R X term. It's negative. 60. I want half of that, which is negative. 30. And when I square it, that's what I'm gonna put back up into my equation. Plus 900. No, I can't just add 900 so I need to subtract 900 as well. That means I haven't made any fundamental changes to my equation. So writing that here I get X squared minus 60 x plus 900 that's my perfect square. This piece I need to move outside of my parentheses, remembering that everything in the parentheses is being multiplied by negative 20. So when I pull it out, I am actually adding 18,000, which I'll combine with the 32,000 that's already out here. The final form of this function then is going to be negative 20 times X minus 30 square. There's my perfect square. I was going for plus 50,000. This is the Vertex. I can pull it right off of this form. The Vertex is going to be at the 0.30 50,000. So what does this number actually mean? Remember, this is my ex, So that means I need to have 30 increases of $20. 30 increases of $20 gives me an increase of $600 over my initial starting point of 400. So that means I'm going to be needing to charge $1000 per apartment, leaving me by maximum possible revenue of $50,000 total.

In this problem. We're told about a company that currently has 640 employees and the number of employees has been decreasing by 5% and that we're told that that will continue. And that's gonna continue for the next 10 years, whereas to find out how many employees will the company have after those 10 years? So to do that, we're gonna use the exponential DK formula which says see times one minus R to the power of X. So what we have to do is be able to read that problem and understand what value corresponds to which variable. So C is your initial value. So they were told that there's currently 640 employees. One minus are are is your d K factor, your rate of decay, which is 5%. We're gonna write that as the decimal 0.5 and X is the term. How long is this happening? This is gonna happen for 10 years. So we're gonna write our exponents of 10. So now we're gonna go ahead and simplify inside the parentheses will do that. Subtraction. One minus 25 when 95 power of 10. And that will evaluate to 383 0.192 And that despot was on for a long time. I rounded it, but because we're talking about employees, we're talking about people. It makes a lot of sense to around this to the nearest whole number. And so we're just gonna write 383 employees.


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