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A tirm produces two types of sugar; A and B at a constant average cost ofRM 2 and RM 3 per kilogram: respectively. The quantities 4A and 9B (in kilograms) of A and ...

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A tirm produces two types of sugar; A and B at a constant average cost ofRM 2 and RM 3 per kilogram: respectively. The quantities 4A and 9B (in kilograms) of A and B that can be sold each week are given by the joint-demand functions QA 400(P8 Pa and QB 400(9 + Pa 2PB) Pa and Pg are the selling prices in dollars per kilogram of A and B respectivelvOhDetermine the prices of A and B at which the firm can maximize profit: 6 marks)

A tirm produces two types of sugar; A and B at a constant average cost ofRM 2 and RM 3 per kilogram: respectively. The quantities 4A and 9B (in kilograms) of A and B that can be sold each week are given by the joint-demand functions QA 400(P8 Pa and QB 400(9 + Pa 2PB) Pa and Pg are the selling prices in dollars per kilogram of A and B respectivelv Oh Determine the prices of A and B at which the firm can maximize profit: 6 marks)



Answers

A candy company has 150 $\mathrm{kg}$ of chocolate-covered nuts and 90 $\mathrm{kg}$ of chocolate-covered raisins to be sold as two different mixes. One mix will contain half nuts and half raisins and will sell for $\$ 7$ per kg. The other mix will contain 3$/ 4$ nuts and 1$/ 4$ raisins and will sell for $\$ 9.50$ per kg.
a. How many kilograms of each mix should the company prepare for the maximum revenue? Find the maximum revenue.
b. The company raises the price of the second mix to $\$ 11$ per kg. each mix should the company prepare for the maximum revenue? Find the maximum revenue.

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In this question were given the weekly costs of producing extensive sugars believe is this? So we'll let C. X. Be the weekly cost of producing extensive sugar. And see X. Is given by these equations row 0.1 X. Square plus five X plus two two 50 And that's in dollars. Now our job is to find the level of production for which the average cost is minimum. Now average cost. Well let average cost the why. And this is actually see X over X. So my wife is equals two. If I were to divide the sea X over X. This is what I get 0.1 x. Last five plus 2 to +50 over X. Oh I can write the 0.1 x plus five plus 2 to +50 X four minus one. Now I want to find dy dx let's differentiate. Okay now since these are three terms added together we can do term by term defense station For the first time. 0.1 x. When a different shape I get 0.1 plus when five is a constant I differentiate I get zero plus now 2 to +50 Expo minus one lifted 2 to +50 At one side differentiate extra power minus one. Open the bracket, bring down the power minus one, repeat the X. Subtract one from the power so minus two. So this is what I get. My dy dx Is equal to 0.1 -2-50 over X square. Now we want to find the average cost of minimum means we need to set this dy dx 20 Mhm. So when I set this to zero I get 3.1 equals 2 to 50 over X. Square. So solving it, my ex is square root off 2- 50. Or was your .1? And that would be 150. Yeah so it X. equals 250. My average cost its minimum. Okay now for the second part of this question do you want to show that the average cost equals to marginal cost? So at this value so at x equals 250 my average cost Is a c. over 1 50. So I'm going to some 150 into every X. I see here. So here we go. This is 0.150 square Plus 550 Plus 2-. 5.0 Over 150. And this will start to be 35. Now marginal costs. Yeah margie Nikos he's actually see prime X. Yeah. So at this value 150. So I'm just gonna put in 100 and 50 into my C. Prime now to find see prime. I need to differentiate over here. So C crime as I'm running out of space I'm just gonna do it here. See Prime. Excellent. I'm just going to differentiate 0.1 X Square. I will get 0.1 times to freeze the x. Power -1. And that's just the these Plus 5X. Where differentiate? I'll just get five Plus 2-50 when differentiates just going to be zero. So my C. Prime It's just gonna be 0.2 x Plus five. So I'm just gonna set up 150 Into the X. over here. So is this 3.2 Come 150 Plus five. This works out to be 35. So we have shown that they are the same. The average cost is the same as the marginal cost. At this level. Production of x equals 250 tons.

Hello. This problem is above a company that produces and sells shirts were given this faras the cost is concerned that the fixed costs are 7000. Those on the the variable costs are, uh, $5 per ships. All right, um, stars revenue. Well, the room you side we have in part a were given that he is equal. The process selling is so there's per ship on in a We need to find the cost of revenue function ser que on are you right now with start assuming that the cost function is a linear function, we have The Formby bus came here where b represents the fixed costs on dame represents the variable growth. So see, if you equals 7000 plus five years. Okay. The revenue revenues, defined as the price per unit times the quantity sold to prosper unit in parties 12. So we have our Q b equals in part B birth B. We have that the man function, the demand function. The market mouth function is given by q equals 2000 minus 40 p on and the peace. The president s Q. Is the quantity off shirts? No. When P equals Fogel's pressure. They want to know how much Q is so. What we do is we insert key instead of BBC talk sort of. Q equals 2000 minus 14 times. Well, which equals the month Ford, the former baby, which is 1520. So what profits? Profits Profit is defined as the difference between the revenue and the cost. So we find the prophet ad 15 20 is the revenue have 15 20 minus the cost after being the revenue function is given a storm. Q. So this will be 12 times 15 21. That's probably 20. And the cost is given by 7000 plus five times, which is 7000 plus five times uh, 5 to 0, about our calculators way have profit is 3600 on for the North. That souls Probably good part scene with you. The demand equation. This is the model equation to express the cost and revenue as the functions off price P. So we substitute que in both of these real 2000 ones. 40. So let's see the cost off P is going to be 7000 plus five times. Never substitute que with 2000 2000 Minus 40 being 7000. My, uh, Too hard. Thank you. Use us The cost function in terms of P prices 17,000 minus 200. You know, right construction in terms of B on the revenue in terms of P is going to be big times. We substitute que 12 your school. Well, let's be 2000. My 40 b uh, was given for part a part C. We're generally Fernie Press looking for the rest search. So this would be 2000 B minus 40 b squared. This will be in terms of press the revenue and we now find profits off key to be full to the revenue off be waas The cost in firms must be which this march this 2000 B minus 40 p squared March 17. Well, so you pee. Uh, no one is a little Take down the parentheses for B squared. My 1000 plus 200 p rooming. We have minus 40 b squared or to be square. He's 2200 peas minus 17. Like this is ah profit function In terms of people there we have with see for the we need the graph meaning? The breath. Profits against press. Okay, so this graph was made in this most groom, uh, y represents the the profits. Uh, this represents the press. Okay, so the maximum point, this is a parable of turns down because the leading coefficient is miles 43 which means that ah, a quadratic function with the negative leading coefficient. He's has a problem. The turned that opens down. So this here this verdict is the maximum points. The maximum profit is maximum profit is $13,250. When the company charges when the company the charges, Uh, 20 simple city, you should Okay, um, this covers the whole problem. Uh huh. But I have a nice day.

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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