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Q.3. A multimedia company produces DVDs. The management decided to charge Omani Rial 50 per DVD to sold to retailer. The total cost C(x) in Omani Rial of produc...

Question

Q.3. A multimedia company produces DVDs. The management decided to charge Omani Rial 50 per DVD to sold to retailer. The total cost C(x) in Omani Rial of producing and selling x units is given by (12 Marks) Find the belowa) How Many DVDs must be sold by the company for break even? (04 Marks)b) How Many DVDs must be sold to produce a profit of OR 1400? (04 Marks)c) What is the profit if 100 DV

Q.3. A multimedia company produces DVDs. The management decided to charge Omani Rial 50 per DVD to sold to retailer. The total cost C(x) in Omani Rial of producing and selling x units is given by (12 Marks) Find the belowa) How Many DVDs must be sold by the company for break even? (04 Marks)b) How Many DVDs must be sold to produce a profit of OR 1400? (04 Marks)c) What is the profit if 100 DVDs are sold? (04 Marks)Rubrics:a) 1 mark for correct formula, 02 x 0.5 mark = 01mark0.5 mark for each correct step, 06 x 0.5 mark = 03 marksb) 0.5 mark for correct formula, 02 x 0.5 mark = 01mark 0.5 mark for each correct step, 8 x 0.5 mark = 04 marks c) 0.5 mark for correct formula, 0.5 x 01 mark = 0.5mark 0.5 mark for each correct step, 7 x 0.5 mark = 3.5 marks Solution:



Answers

Based on market research, a film production company in Ectenia obtains the following information about the demand and production costs of its new DVD:
Demand: $P = 1,000 - 10Q$
Total Revenue: $TR = 1,000Q - 10Q^2$
Marginal Revenue: $MR = 1,000 - 20Q$
Marginal Cost: $MC = 100 + 10Q$,
where $Q$ indicates the number of copies sold and $P$ is the price in Ectenian dollars.
a. Find the price and quantity that maximize the company's profit.
b. Find the price and quantity that would maximize social welfare.
c. Calculate the deadweight loss from monopoly.
d. Suppose, in addition to the costs above, the director of the film has to be paid. The company is
considering four options:
i. a flat fee of 2,000 Ectenian dollars.
ii. 50 percent of the profits.
iii. 150 Ectenian dollars per unit sold.
iv. 50 percent of the revenue.
For each option, calculate the profit-maximizing
price and quantity. Which, if any, of these compensation
schemes would alter the deadweight loss from
monopoly? Explain.

So we have a couple of equations. We have a revenue equation for these wrist bands per month and that 75 x minus 0.2 x squared. And we have a cost equation. Her month based on the number of wristband sold, and that's 32 x lost 1750. So that's our cost based on how many they sell. And we want to find the Max revenue and we can find that a number of different ways And one way is to remember that this is a quadratic that opens downward. And if we have a quadratic that opens downward, we know that the access of cemetery will take place at negative, be over to a so we can we want to rewrite that quadratic. Well, I won't totally regret it, but this is our little A. This is our little B and then r C is zero. So we'll have are the opposite of B over two times a, which is negative 20.4, and if we take 75 divided by 750.4, we get 187.5, so we could say either 187 or 188 wrist bands is the number that they would sell on. How much profit would they make? Let's just go with the lower number. So let's plug that in there. If I hit 75 times 1 87 and then subtract away 0.2 times 1 87 squared, that ends up telling me about $7031. I get in 20 cents if I use that one. So so there's our maximum revenue. Now we go to Part B and we talk about profit. And we know that the prophet would end up being the amount of money they take in revenue minus the cost. And so we can subtract those two models. So we have that 75 x minus 0.2 x squared. And we need to subtract away that cost model 32 x plus 1750. And why don't we just run this negative through like so? And then we can add up are like terms, and when we dio, we've got that negative 0.2 x squared in the front. We can combine these two X terms to get plus 43 x and then we're gonna have this last term, which is minus 1750 and that gives us a profit equation. And again, that's a quadratic that is opening downward. And so then we want to find what that maximum profit is. And once again, we want the max profit. It's an app right there we can take and find the opposite of Be over to a So we have the opposite of B over to a which again, this negative point four. And if we take 43 divide that by negative 430.4, we're gonna end up getting 107.5. So we'd say that that maximum profit is after they sell 108 or 109 depending on how you're going around to get that. We can't get it to be exactly at that spot with 3.5. And what is that profit? Well, let's go back to this equation and I'm gonna take 107 And I'm gonna store that as X and then I'm just going to type this in as negative 0.2 and hit x squared plus 43 x minus 1750. And I find out that that, um maximum whoops. I don't think I hit something quite right. Andi. I only get a maximum profit of $561.20 if I just typed it incorrectly. So, uh, now there may have been a typo up here and that the question show that I'm not seeing But that's what I get from my answer. Based on what my question is printed from from New Marais did so hopefully.

So here we're going to be doing some profit analysis. Um And we want to write a total cost function. So c. Is going to be equal to okay $50 per unit. Um and the initial investment was 350 1000. And the revenue Is equal to make $120 per unit, 61 20 x. So we're going to make these X. Values from -5 to about 100. Uh huh. So zooming out. We can now look at the cost and the revenue function, see where they meet up and we see that that's at about 5000. Um So the Prophet as we see is equal to revenue minus cost, which tells us that they break even at five and then they'll end up having a profit. Um once we have 13,000. So x equals 13,000. We see that at 13,000 will have a profit about $550,000. And they broke even at 5000.

Okay. This is problem number 79 in which, if x is the number off DVDs number of devotees produced then that every new function is given should be given. But has found that revenue from the sales off David eases $55 per deputy less Sales costs off 100. So revenue function must be There's a new function. Must be five x minus 100. All things should be in dollars. Thank you.

Okay, So we'll be looking at some problems that combine both linear and quadratic functions, particularly with respect to manufacturing. And the 1st, 1st example we're gonna look at has to do with uh, wristwatches. So a company is selling making and selling wristwatches and there's what's called a revenue function. That just means the amount of money that they make on the rift, not profit, but the amount of money that they bring in the income relative to the risk, the wrist watches. And that revenue function is 75 x minus 2/10 X squared. And then we also have a cost function that is the cost to make the wrist watches. And so when we look at those two graphs together, then the the if we're looking first of all, let's just look just at the revenue function. So if we want to find out how many wristwatches we need to sell in order to maximize revenue without any regard to cost, then we're gonna maximize revenue right here At 187.5. So obviously we can't sell half a wristwatch. So our revenue would be maximized at either 187 wristwatches or 188 wristwatches. Both of those would give us maximum revenue. So it's gonna max out at 187 or 188 revenue maxes out At 187 Or 1 88. And that maximum revenue would be whatever value we get when we substitute 187 or 188 into The function. So it would be, we're gonna take 75 multiply that by 187 -2/10 Times 187. And that gives us a maximum revenue of $13,987 and 60 cents. Oh no, that's not right, I didn't square that. It's 75 times 187 minus 2/10 Times 187 Squared. Here we go. That's better. Um and that's a maximum revenue of $7,031.20. I knew that it wasn't right because it didn't agree with. There wasn't close to that Vertex maximum there. So when we sell 187 or 188 Wrist watches, we have an income, just an income, not a profit of an income of $7.31, 20 cents. Now, the profit function, if you think about this realistically a profit function is just gonna be the cost that it takes to the revenue, the total that you bring in for the sale of whatever product or selling minus the cost to make that product and that's going to be called the revenue. The profit function. So the profit function is going to be simply the revenue, the amount that you bring in minus whatever it costs you to make that product. So when we substitute These in, we get a profit function of 75 x minus two tents, x squared- your cost function of 32 x plus 1750. And then when you distribute that negative sign, you get a profit function, you distribute the negative sign and you re arrange your variables to make to write your expression in standard form, you get negative 2/10 X squared plus 43 X -1 750. So that's going to be your profit function. And then we'll look at our profit function in order to maximize profit. So then when we graph the profit function, it's gonna look like this here, this purple graph And clearly we're gonna have a maximized profit here at 107 five wristwatches. So profit is maximized on either side of that profit is maxed At 107 or 108 watches. And that maximum profit Is going to be. So we're just gonna take p of 107 and that gives us negative point to times 17 squared Plus 43 times 1 7 -1750. And that gives us a profit of $561.20. Now notice it's not exactly 561 25 because that's if we were just at 107 and a half watches, which we clearly can't do. Um so your maximum profit is also going to be the same at 108 because remember that axis of symmetry of your parabola goes straight down the middle. So these two points on either side of the vertex are going to be um reflections of each other across that That axis of symmetry. So your profit is maximized at 107 or 108 watches for a maximum profit of $560.20. And then if you think about it, um why is the profit max? Why are these two answers different? So why is the revenue max? And the prophet max is different? Well, clearly it's because the revenue max doesn't take into account the cost to produce the material or to produce the product. In this case the wristwatch.


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