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A company can manufacture three types of ladders. Profit is P100per ladder of type A, P150 per ladder of type B, and P180 perladder of type C. Each ladder must be p...

Question

A company can manufacture three types of ladders. Profit is P100per ladder of type A, P150 per ladder of type B, and P180 perladder of type C. Each ladder must be processed through threecenters according to the following requirements:Minutes in Minutes in Minutes inCenter 1 center 2 center 3Type A 5 8 5Type B 5 8 10Type C 5 8 10minutes available 560 880 900Determine the number of ladders of each type that will maximize theprofit.

A company can manufacture three types of ladders. Profit is P100 per ladder of type A, P150 per ladder of type B, and P180 per ladder of type C. Each ladder must be processed through three centers according to the following requirements: Minutes in Minutes in Minutes in Center 1 center 2 center 3 Type A 5 8 5 Type B 5 8 10 Type C 5 8 10 minutes available 560 880 900 Determine the number of ladders of each type that will maximize the profit.



Answers

PROFIT Refer to Problem 89 . The cost of manufacturing 8,000 BTU window air conditioners is given by $$ C(x)=10,000+90 x $$ where $C(x)$ is the total cost in dollars of producing $x$ air conditioners. (A) Find the profit function. (B) Find the number of air conditioners that must be sold to maximize the profit, the corresponding price to the nearest dollar, and the maximum profit to the nearest dollar.

For this problem, we are told that a firm produces two types of earphones per year. X 1000 of type A and 1000 of type B. If the revenue and cost equations for the year are in millions of dollars are of X. Y equals two X plus three Y. And C. Of X Y equals x squared minus two xy plus two Y squared plus six X minus nine Y plus five. We are asked to determine how many of each type of earphone should be produced per year to maximize profit. In addition, we are asked what is the maximum profit? So the Prophet as a function of X. Y. Which I'm going to write as P there. It's going to be the revenue two x plus three y minus the cost. So we have two X plus three Y minus X squared minus two xy or rather plus two xy minus two Y squared. And we have that we'd get the two X -6 X. So that will become negative four X. Out front. And we have three y minus or plus nine Y. It will turn into 12 Y. And then we'll have a -5 to end things off there. So we want to minimize this, which means that we want to look For the point where the partial derivatives with respect to X&Y both equal zero. So the partial derivative with respect to X is going to be negative for minus two, X plus two, Y equals zero. In other words, we can rearrange this to get why is going to equal X-plus two. Yeah. Then we can take the partial derivative with respect to why it will give us 12 plus two x minus four Y. We want that to equal zero. We can then substitute in Y equals X plus two. So that turns into 12 plus two, X minus four, X minus eight equals zero. So let's see here, we'll have 12 -8. So that's going to be paused at four two X minus four. X is going to give us negative two. X. You get four minus two. X equals zero. Or that X is going to have to equal to Which means then that why is going to equal for? We want to confirm that this is going to be a maximum by taking the second partial derivatives. 2nd partial derivative with respect to X is going to be negative too. The mixed derivative, it's going to be positive too. The second partial derivative with respect to why It's going to be -4. So our a c minus B squared. It's going to be negative two times negative four. So that's going to be positive eight minus two squared. So minus four equals just four which is greater than zero. And we have a second derivative with respect to X, which is less than zero, which is going to indicate that that is in fact a local maximum for profit. So we have that the .24 is a maximum, And we want lastly to calculate out what the profit will be at that .24, I'm going to pause and calculate that offscreen. So the value of the prophet adds the .24 is going to be 15, so the maximum profit is going to be $15,000.

So part A of this problem asked us to find a function that models the weekly profit in terms of price for Peter. So we do. That is by letting X is equal to the number off, um, number off $1 increases. Um, and eso our profit will be our revenue minus cost on a revenue is the price times quantity. So the price would be, ah, the base price, which is $10. Uh, plus, ex witches are $1 increases. They'll be our price per feeder on our quantity off eater is, um, as we start at 20 per week and since for every dollar Ah, system, since for every dollar increase we lose to, ah to cells per week, I'll be minus two X. So the 10 plus X represents our price and 20 minus two x represents our, um, our quantity. And, uh, so if we just expand its outs ah, frees foil. First term is 200. Uh, our terms are negative. 2020 x in our terms are close 20 x and last terms or minus two X squared. So this models are, um, yes. So this models are, uh oh, our revenue So now we gotta factor in the cost to find a total profit because profit is equals revenue by its cost. So the cost is equal to the cost per feeder times the number of features. So the cost per feeder is $6 multiplied by the number of futures which he found above is that that's the same things. The quantity. So it's 20 um, honest to x. And, um, that is equal to if you expand its up ho 1 20 uh, minus 12 x. So if we practice two equations because this is a revenue eso cabos couple are and our cost is couple see and profit is equal to R minus c. It is equal to ah 200 minus 20 exposed 20 extra zeros which will 200 minus two x squared A minus 1 20 Oh, plus 12 x another. We see we have ah quadratic formula. So our quarterback formula is equal to negative too. X square, close toe X plus 80. Now we can maximize this, um dysfunction. So this is that's the answer to problem, eh? Is that Ah, we found a function that models Ah, the weekly profit in terms of price for future now, part B s us. To what price should the society charge for use fear to maximize profits. And what is the maximum weekly profit? So, essentially, we had to find a maximum its function. And when we do, that is likely in the square. So you're taking Nate id Teoh. We put in standard form and defy the Vertex since is negative. You know, the Vertex opens downwards, so the problem would happen. Sorry where the problem is downwards. So the vertex would be the maximum. So, um, we take negative too common and left the X squared off my six X. And with this, we'd love to square it says plus nine. And, uh, since we're adding native 18 to this equation, the most attractive 18th this question so was 18 of plus 80. So that is, um, X plus three. I swear, uh, close 90 hit. So, uh, for X minus three. Uh, so we must have X is equal to ah three. And we know that three is our access equal to the number $1 increases. So we must increase our price by $3. Um, so if you increase our price for $3. The price per future becomes 13 because 10 close three is 13 s 0 13 is the price of the society's just charge for each Peter. And what is the maximum weekly profit? That's just 13. Since we plug this back into the original equation, right? And, uh, people it is back in this equation were end with 98 so we ended with $98 total.

Okay. Company makes television sets, and the monthly cost of production is $72,000. 72,000 plus 60 x dollars x is the number of TV sets they produce on the price demand equation is P equals 200 minus X over 30. And that's for X between zero and 6000. So we want to find the maximum revenue and then on the second one, fined the maximum profit production level and the price. And then in the third one same question only add $5 of tax to each one. All right, so first, maximum revenue. So the revenue would be the number of sets that you produce times the price of the sets to the 200 minus x over 30. So if you multiply that out, that's 200 X minus 1/30 X squared. And the reason I wrote that is because we want to find the maximum. And to find the maximum of anything, we have to take the derivative and set it equal to zero. So I wanted to make sure you know how to take the derivative of X over 30. Okay, so our prime is the derivative of 200 X, which is 200 minus 1/30 times the derivative of X Square, which is two X and that's equal to zero. That's how do you find the maximum? So that's 200 equals two X over 30. Okay, so cross multiply, So 6000 equals two x So x is 3000. So we're gonna make 3000 televisions, and so then the the revenue will be 200 times 3000 minus 1/30 times, 3000 squared. You know what? This would be easier if I would put it in the factored up version. I wanna have to get out my calculator. Okay. Into this for into this version right here. So it's 3000 times 200 minus 3000 over 30 3000 over 30. That's 300 over three. That's 100. So that's 3000 times 200 minus 100 which is 100. That's three with five zeroes, five zeroes. So 300,000 is the revenue. Maximum revenue can now be fined the maximum profit the production level to reach the maximum profit and the price to charge for the maximum profit. So the maximum profit is different from the maximum revenue. Revenue is just how much money you make. Profit will be how much money you make. Minus how much you call, how much you spent. Sorry. All right, so be profit equals revenue minus cost. So x Times 200 minus x over 30 minus the cost, which was 72,000 plus 60 x. Sure, 60 x. Okay, so let's see if we got some, like, terms we can combine before we take the derivative. So I get minus 1 30 x squared plus 1 40 x minus 72,000. So that's the prophet equation. So I'm gonna take the derivative because they want us to find the maximum minus 1 30 times two x plus 1 40 equals X. So that gives me two. X over 30 equals 1 40. So X equals 1 40 times 30 over to 21 00 So the mac make the maximum profit instead of making 3000, which gave us maximum revenue. We only have to make 2100. So what will the prophet be? Well, it will be. We gotta plug it in minus 1. 30 times 2100 squared, plus 1 40 times 2100 minus 7,272,000. Okay, I'm getting out my calculator now. Okay, 21 00 squared. Divided by 30. So equals minus. Just sick. Cat problem here minus. Oh my Sorry. 147000 plus 1 40 times 2100 to plus 2 +94000 minus 7 to 000 equals. Yeah, for some chiro. So to your general 75,000. All right. Okay. So what? I better go. Okay. Did I answer the question? Fined the maximum revenue. Yes. Um, be fined the maximum profit. The maximum profit is this. What's the production level? This. And what's the other question? What? The price to charge So P Waas 200 minus x over 30. So 200 minus 20. 100 over 30. That's 200 minus 70. So 130 max Profit production level price to charge. Alright, car. See, the government decides to tax each set $5 and now we want to answer part. Beat questions again. Okay, So what does that change up here? Well, it will change the price demand equation. It will be, um X is the number of TV sets produced? This is how much we're gonna charge for them. Plus, now we have to tax each set $5 so we're gonna have to take $5 away from our profit per set. Okay. So part seek, uh, the monthly cost is still the same while 60 x plus 60 x KP 200 minus x over 30. That's the price of one television set, minus $5. Now that we're not gonna get you know what? That's don't put it there. Let's write the same equation we had before. Profit profit will be 7200. No, no, no profit will be X times the number of television sets times the price per television set minus the cost also minus the tax, which is $5 times how maney TV sets. We sell five X. So instead of putting it there, although it would have been the same, it makes more sense to me. This is number of TV sets times price per TV set. That's the revenue minus the cost minus the tax. That's 200 X minus 1/30 X squared minus 7200, minus 60 X minus five x so minus 1/30 x squared I want minus 1 35 x minus 7200 72,000. Plus I mean here because it's 200 minus 16 minus five. Okay, so the derivative one. 30th Time to X and 1/15 x folks, that's minus. That's that minus right there minus 1/15 x plus 1. 35 said it equal to zero. So 1 35 equals 1/15 x So x equals 1 35 times 15. So 2025 is the production level. The prophet is minus 1/30 times 2025 squared plus 1 35 times 2025 minus 72,000 2025 squared. Divide by 30. Make that negative. Plus, uh, there I got minus 136687.50 plus 1, 30 votes. 1. 35 times. 2025. 1 35 times. 2025 273375 morning of 72,000. So minus right now. This 1366 7, 50. Okay, so the Prophet iss 64 687 50 za max profit. And then what's the price we should charge. So the price this one was a big P. This little P price. Switzer's charged 200 minus 20 25/30. When you're 25 2025 divided by 30. Minus 200 gives me 132.5. So, uh, price per TV 130 to 50.

Anyone in this percent past mere mortal up over it once before didn't give you an informant. And then you find out that maximum on the I don't know if I know that. What I would be excellent bought that you didn't party. So party bottom window didn't find up over something. There's a groper Cantonese. Can I? What city is it? But this is honestly one party, Ok? Minus. Yes. What? They're going got experiment. Not in prison. Don't know on their musical sell part of it. So it means you and your ex will be their central beginning. And don't defective products will be their faces. Indolence older to do it within less X OK minus four. Found Cellulosa Parexel forces do in six like this in device there? Yeah, I conveyed. X is not next in the line. So corporate funds and Lardner getting here when being minus two legs And do invite a sexy school for less excellent simplified people. My Vanya vicinity. My message by Excel City development is the eggs find that school exist lives and this device on them. What I'm there is a mother who works a spread on the one d minus. It is bloodless. So this is the response of all noble maximize. Well, part of the offensive. No, Y my next Olympic gold one. It will be a to spread minus sex. My nest 40 Now. Next, what I can do minus go and do this. And I'm like people tech spread. So I stand by 66 Bless my name rd minus nine numbering minus 40. So I pulled up on verify simplified and minus going toe. This is X minus 10 the holy square I convey. It's Linus didn't hold you. Square nexus minus story. Angel, this is minus nine minus 40 minus 49. Minus minus Place so pleasant to 4900. So profits before doing. Finally, I'm getting it done, Mr. Under formal for idiots prison minus boy and girl X minus city on display. Unless we're in 49 which inside you. So it's authentic. You will maximize up this part of the you know. So then this part with you know the next portal? Well, we can bite varieties except for the face off me. So this is that place. It's a book that, you know little maximize up over till next month authorities. You makes it vocal. It's far too dio Well, 90. So this is not Mexican brokered nine year doughnut, and, uh, I don't buy the unities. It's for Duke University is not the answer. Thank you.


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