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Cehack Industries needs design Lalk with volume of 3T In" Fheir materials Cngincer; Sochic Hejm_ estimates it will cost $15 per I ? to build the Lube portion...

Question

Cehack Industries needs design Lalk with volume of 3T In" Fheir materials Cngincer; Sochic Hejm_ estimates it will cost $15 per I ? to build the Lube portion and 824 pCr I? for the CapS. Whal is Ghe lowest possible cost to Iafacture the tank and what are Lhe corresponding ditensions? For complete crecit YOu InusL include all of thc following with_Gxplaatious:Determine what quaLutity is to byc oplimized (the problem is to Ialximize / Ininimize what?) Delerminc single variable fiuction that

Cehack Industries needs design Lalk with volume of 3T In" Fheir materials Cngincer; Sochic Hejm_ estimates it will cost $15 per I ? to build the Lube portion and 824 pCr I? for the CapS. Whal is Ghe lowest possible cost to Iafacture the tank and what are Lhe corresponding ditensions? For complete crecit YOu InusL include all of thc following with_Gxplaatious: Determine what quaLutity is to byc oplimized (the problem is to Ialximize / Ininimize what?) Delerminc single variable fiuction that describes this quautity Delermin a appropriate domain for Lhis fuuction Determine the extrema by using aH apprOpriate extreIe valule test [rom caleuls: (Options: closed interval IeGhod_ first derivalive tesl_ Or SC'COIc derivalive test.) In other words; YOIL InuSL make sure to verily that YOn have found the extrema, Ilot just a critical point _ Answer the question using complete sentence and appropriate uits. To aid in COHHIIicaLion; pleatse rOud results to Uhc Second decimall place



Answers

Maximum Profit The cost per unit in the
production of an MP3 player is $\$ 60 .$ The manufacturer
charges $\$ 90$ per unit for orders of 100 or less. To
encourage large orders, the manufacturer reduces the charge by $\$ 0.15$ per MP3 player for each unit ordered in
excess of 100 (for example, there would be a charge of
$\$ 87$ per $M P 3$ player for an order size of 120 )
(a) The table shows the profits $P($ in dollars for various
numbers of units ordered, $x .$ Use the table to
estimate the maximum profit.
$$
\begin{array}{|c|c|c|c|c|c|}\hline \text { Units, } & {130} & {140} & {150} & {160} & {170} \\ \hline \text { Profit, P} & {3315} & {3360} & {3375} & {3360} & {3315} \\ \hline\end{array}
$$
(b) Plot the points $(x, P)$ from the table in part (a). Does
the relation defined by the ordered pairs represent $P$
as a function of $x ?$
Given that $P$ is a function of $x,$ write the function
and determine its domain. (Note: $P=R-C$ ,
where $R$ is revenue and $C$ is cost.

Okay, guys, this is Chapter 15 problem. Tang and this question were given the following equations the demand equation to 11 years. Barbara Revenue and Marginal cost equations. After the first part of questions were asked, What's the profit? Maximizing price in quantity? Recall. The profit maximization occurs where marginal cost equals a marginal revenue. And so here we set the two equations equal to each other. Marginal revenue is 1000 minus 20 hugh and marginal cost of 100 plus 10 Q. And so we get 900 equals 30 Q and so 30 is going. T equals. Q. That's the quantity. To get the price, we plug this into the demand equation, where price equals 1000 minus 10. Q. The price equals 1000 minus 10 times 30 Ah, which is 300 and sale price equals 700. The second part of the question is asking us then what is the welfare maximizing price in quantity So wealthier maximization occurs where marginal cost equals demand. And so we were given those two equations in the first part of the problem. And then we said those eagles to each other. So one original cost is 100 plus 10. Q. In this equals 1000 minus 10 Q. There we get 20 Q equals and 900. We're q equals 45 and we give the price by planets into the demand equation. Where we have 1000 price equals 1000 minus 10. Q Price. People's one of founders and minus turn times 45 which is 450 price equals 560. For the third part of the question, we're asked to calculate dead weight loss and Sarah remember that dead weight loss Here's the graph for a monopoly that monopoly is going, said he was marginal revenue, the man's hand. Marshall cons they had The monopolist is going to produce where marginal running with marginal cost. But they're going to charge this piece dollar and vest. Right here is the dead weight loss. We want to catch me carrying that triangle different parts and Kirby. We saw that the difference between the welfare maximizing profit maximizing levels of production in the prices was a difference between 40 45 30 for the quantity in between 705 150 for the press and said with a triangle Therefore, we have to cut it in half. It was gonna be 1/2 times 45 minus 30 times 700 minus 550. That's gonna be 1/2 times 15 times 150. And that's going to equal 1125. The last part of the question. Then out then tells us that the company is considering killing the director of this movie that they're producing one of the four options. Either flat rate of 2000 2000 50% of the profits. 150 Persian. It sold for 50% of the revenue. T to tell this to calculate the new profit maximizing person quantity and then asked which one alters the dead weight loss. So this is actually easier than it seems. The only one of these that's goingto altar. The profit maximization is going to be this number three because that's the only one that alters either the marginal con for the marginal revenue. All the other ones were dealing with the totals. And so the new marginal cost curve I was in the old marginal cost curve wass um 100 plus time acu. So they need a margin. Call it new. Marginal cost is going to be 250 plus 10. Q. Then we just do exactly what we did before we set a marginal cost equal to marginal revenue minus 20 Q. And we saw for Kyu Won. This is going Q equals 750 peoples 25 fun that back there and demand equation to have price equals 1000 minus 10. Q. But she goes 1000 minus turn times 25 on This, of course, is 257 year price people 750 is that there is only one of these has also goingto alter the boss. Is this numbers this third option? So this is chaplain.

So for this problem, they give you a set of data to plot, and I'm gonna not gonna plot that. But we're gonna come up with the model. And so, uh, so we do have profit is a function of X. That is true. And X is the number of units sold, number of units solved. And they tell us that if the number of units sold is less than 100. So if X is less than or equal to 100 we know that it costs them. Uh, costs $60 to produce one of these units, and they sell it for $90. So they're gonna end up. If I look at their profit for this setting, we know that their profit will end up being the 90 times how much they bring in. So they're gonna bring in $90 for each item. However, it cost them 60 times. Uh, X is their cost. So this is how much it will. They'll bring in. So the revenue and then this is minus the cost. And then that's gonna end up being their profit. So we know that that simplifies Donald will simplify that in a minute. But what do we do if if we have an object that is more than 100 and if it costs more than 100 then they say they reduced the price. And so we know that, um, normally the price of an object is $90. That's how much they sell it for. But they're going to reduce that fit by 15 cents for every unit over 100. So we need to find out how money over 100. So if we take how many units and we already know the number is bigger than 100 if we're using this part of the piece wise function. And so let's say they were selling 101 objects, then we would need to subtract away 100 from that. And that would be the number of items that are over 100 and they're going to reduce the price by 15 cents for each one of those items over. So if they had to say this was 110 then 10 times they're going to reduce the price 15 cents. So if we put 110 in here 110 minus 100 would be 10, and we'd multiply it by the 15, and they'd reduce the price by a dollar 50. So this is what the selling prices but we need to sell. Multiply it by X, and that's going to end up giving. This is our revenue well, and I need small room. So let's erase this guy and we'll move him over a little bit because I leave myself enough room. Here we go, so we'll go X is greater than 100. So this is our revenue when you need to attract away the cost, will. The cost is always staying $60 for each item. So now what do we need to do? We need to clean these up, and we've We've satisfied actually what they asked us to do to write a function for this in terms of the number of units sold. But we can clean this up, So let's do that and subtracting these two, we would get 30 acts. That's what the profits going to be as long as you have something that is from zero to 100. Yeah, and then let's clean this up. We'll clean this up just over here, so that I don't have Thio mess with this down here. So we would be distributing this through so we would have 90 minus 0.15 x and then a negative times a negative is positive. And that 0.15 times 100 would move the death, the place I'd become 15 on then that needs to be multiplied by X and then we need to attract away 60 X. All right, so we can combine these two together and those two ed upto 105 and then we're going to distribute the X. We have 105 Oops, 105 x and then x times. This gives me minus 0.15 x squared and then we're subtracting away 60 x. So now we can combine these two together and then I'll write this final answer down here. So we have negative 20.15 x negative 0.15 x squared. And then again combining these two X values together, we're gonna get plus 45 X and that's going to be when X is bigger than 100. And so really, we have whole numbers for our domain. Yeah, because we can't have negative numbers and you can't sell portions of units. So really, our whole numbers would end up being our domain. And this is a function. And you could even check this function by going in and plugging in 1, 10, 1, 21 30 as the table shows to verify that this will will actually work. Now there may be a cap on how many, because realistically, they probably just can't make you know billions and billions of these things. So they're going to be restricted on how maney they can actually manufacture. So there's probably a cap on how largest would be. But you definitely do wanna have whole numbers, something where you're not in allowing decimals to come in. It's not just really numbers, but you will need a piece wise function.

For this problem. We're being asked to maximize the profit for the mural manufacturing company, which is making plasma screen television sets. So let's see what our equation is for the profit that this company is going to make. Well, there are two types of television sets that it makes. One is a flex scan, which makes $350 of profit. So I'm just gonna mark here that my ex up one is going to be my flex scan model. And the second kind that it makes is the panoramic. And there's $500 of profit for that one. That's my panoramic in the ceramic. Okay, so this is my profit equation. $350 of profit for each flex scan. $500 of profit for each panoramic. Now, what are my constraints? Well, one constraint we have is the assembly line. Okay. Now, on the assembly line, we're told that the Flex scan requires five hours and the panoramic takes seven hours and at most I've got 3600 hours available on the assembly line. Our second constraint comes at the cabinet shop and there were told that the flex can only takes one hour in the cabinet shop, the panoramic takes too. Okay. And adding all of those up, I have 900 hours total available in the cabinet shop. My third constraint happens in testing and packing like that red. So in the testing and packing area, there were told that the flex scan takes four hours, as does the panoramic. So each one of those takes four hours altogether as 2600 hours. Now, I'm going to simplify this one slightly. I'm going to just divide everything by four just to keep my number's a little smaller. And this becomes 650. I do need to remember, though, that if something comes back with the slack variable for that equation, I need to remember that I made an adjustment. Will have toe, um, work on that at the end if something comes back for that one. Okay, Let's look at how I can add my slack variables so I could enter these into my grid. My first equation, my assembly. I'm just gonna write these in blue, so they kind of stand out a little bit. Five x up one plus seven x up to plus my first slack variable is going to equal 3600. My second equation. I'm going to add my second slack variable, and that will equal 900. And then for my third equation, I have my third slack variable. And that equals 650. Okay, Knowing all of those, I can now put together my grid. I have two X variables. I have three slack variables and I have ze. So let's enter our blue constraint equations in here. If I pull out all of my coefficients, I have 57 and one I don't have any of those variables and it equals 3600. My second equation is 12 It has the second slack variable in that equals 900. And my third one is one and one, and it has my third slack variable equaling 6 50 and my indicator row at the bottom I'm going to take and I'm gonna market with a blue arrow appear at the top, my equation for Z and I'm going to set everything equal to zero. So when I pull all of the terms over to the left hand side, I have negative 350 x of one minus 500 except to no slack variables and dizzy. Hey, so here's my grid. Okay, Now, I know that I'm going to be redoing my grid down here, So I'm going to going to set this up really quick. All of my same variables. Now let's take a look at where our pivot point's going to be. If I look at my indicator row, negative 500 is the biggest negative number I have. So we'll be looking at the ratio of my Constance to the X Up two column. And when I do that, 3 36 100 divided by seven is about 514. Yes, that's got it. With a decimal 900 divided by two is 450 and 6 50 divided by one is obviously 6 50. So the smallest one is the two. There's my pivot point. So the row with the pivot point my second row is not going to change. That stays the same to get rid of my other rose. Well, to get rid of the top row, I've kind of don't have a whole lot of room right there, so I'm just gonna put this off to the side. I'm going to take negative seven times the second row, plus twice the first row. And I'm going to put that back into the first row all the way across in my new grid. And when I do that, I end up with three 02 negative. 700 900. Remember, the goal was to make every number in the same row or same column with the pivot number all equal to zero. Okay, well, what about my third row? Well, in order to get that one that I'm circling here to get that to become a zero, what I'm gonna do is take negative the opposite of the second row, plus twice the third row, and that will go into the third row position. Doing that across the row gives me 100 negative. 1 to 0 and 400. Okay, finally, we need to dio are indicator row at the bottom. To get rid of that, I'll take 500 times the second row plus twice, the fourth wrote. And that's what I'll put in the fourth row all the way across. And when I dio, my output is negative. 200 00 500 02 and then 450,000. Hey, we are not done. We still have a negative in our indicator, wrote negative 200. So to find our pivot point, we're gonna look at the values in the x of one column or Exubera column and compare them to our constants. So no. 1 900 divided by three is 300 900 divided by one is 904 100 divided by one is 400. So this three right here gives me my smallest ratio. That's my new pivot point so and bring that down a bit. So I've got some room toe work and we make our next grid. So the road with the pivot point does not change. So I'm just going to copy Row one all the way across, as is. Hey, I want to get rid of every other number in that except one collar. I want them all to equal zero, so let's look at the second row to get rid of that. I will take the opposite of the first row, plus three times the second row, and that will go into the second row space across the board. When I do that, I end up with 06 Negative to 10. 00 1800 Right. Third row. Well, again, I have the same number in the second row and the third row. So my equation is gonna look the same. Negative are sub one plus three times the third row, and that will go into the third row position. Doing that gives me 00 negative. 2460 and 300. Now we just have to get rid of that negative 200 in the bottom row. To do that, I'm going to take 500 times the first row. I'm sorry. Timmy did write 500. I met, right, 200. Try that again. So 200 times the first row, plus three times the fourth row. And I'm going to put that into the fourth row position. Doing that all the way across that row gives me 00 401 100 06 and one million, 530,000. There are no more negatives in my indicator row, so we are complete so I can start reading the answer off of this grid. Uh huh. Except one just has one non zero. So I could say that three X up one equals 900 or X up one equals 300 except to also has only one non zero. So six X up to equals 1800 or accept to also equals 300. So without looking at any of the other variables, I know that the optimal number of units to make our 300 of each type of television set Now, I can also see that from my slack variables. S sub one and s up to are both going to be zero but s up. Three is not s up. Three. I could say six s up. Three equals 300 or as sub three equals 50. Now, I'm just gonna come back up to the top and remind you s up. Three. If you remember, I made those numbers a little bit smaller. I divided it by four. Everything in my first one divided by four. My second one. So I really need to multiply this by four to go back to the original equation. So s up three for my original equation is 200. So what does that mean? Well, that means that in that third constraint, which was my testing and packing facility, I have 200 unused hours. So I've got some, You know, everything else. I maxed out to capacity, but I still had room available here in that particular department. And what is? We come back so you can see this little bit better. What is my maximum profit? Well, six z equals 1,530,000 or Z equals 200 55,000. That is the maximum profit for selling 300 of each type of television set.

For the given problem, we want to consider this price demand equation and the cost function. So our price demand equation is going to end up giving us P equals negative X over 30 plus 300. The consecration can be 150,000 plus 30 X. Mhm. So what this tells us is we can find P. We already found that. Then we can find marginal cost that's gonna be C. Prime of X, Which just equals 30. And we want to find the revenue function are of X. Is going to equal P. P function time black. So it's gonna be a negative X squared Over 30 plus 300 x. And then we can find our marginal revenue by taking our crime events, which then allows us to find now the profit function and the marginal profit function.


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