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The Audubon Society at Enormous State University (ESU) isplanning its annual fund-raising "Eatathon." The society willcharge students $1.10 per serving of...

Question

The Audubon Society at Enormous State University (ESU) isplanning its annual fund-raising "Eatathon." The society willcharge students $1.10 per serving of pasta. The society estimatesthat the total cost of producing x servings ofpasta at the event will beC(x) = 340 + 0.10x +0.002x2 dollars.(a) Calculate the marginal revenue R'(x)and profit P'(x) functions.R'(x) =P'(x) =(b) Compute the revenue and profit, and also the marginalrevenue and profit, if you have produced

The Audubon Society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students $1.10 per serving of pasta. The society estimates that the total cost of producing x servings of pasta at the event will be C(x) = 340 + 0.10x + 0.002x2 dollars. (a) Calculate the marginal revenue R'(x) and profit P'(x) functions. R'(x) = P'(x) = (b) Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold 200 servings of pasta (in $). revenue$______ profit$ ______ marginal revenue$ per additional plate ________ marginal profit$ per additional plate _____ Interpret the results. The approximate _______ profit loss from the sale of the 201st plate of pasta is $ ____ . (c) For which value of x is the marginal profit zero? x = _____ plates Interpret your answer. The graph of the profit function is a parabola with a vertex at x = ______ , so the loss is at a minimum when you produce and sell _____ plates.



Answers

The Audubon Society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students $1.10 per serving of pasta. The society estimates that the total cost of producing x servings of pasta at the event will be C(x) = 340 + 0.10x + 0.002x2 dollars. (a) Calculate the marginal revenue R'(x) and profit P'(x) functions. R'(x) = P'(x) = (b) Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold 200 servings of pasta (in $). revenue$______ profit$ ______ marginal revenue$
per additional plate ________ marginal profit$
per additional plate _____ Interpret the results. The approximate _______ profit loss from the sale of the 201st plate of pasta is $ ____ . (c) For which value of x is the marginal profit zero? x =
_____ plates Interpret your answer. The graph of the profit function is a parabola with a vertex at x = ______ , so the loss is at a minimum when you produce and sell _____ plates.

So given the function P equals uh 600 minus 0.5 x. We know that the revenue function is going to be x times our uh x times our demand function. So our of acts equals 600 x minus 0.5 X squared. And then um we want to find the profit function so that's going to be the profit function is equal to the revenue function minus the cost function. And we know that the cost function is given to us as zero a lot of zeros to X cubed My 0.03 x squared Plus 400 x plus 80,000. So we end up seeing that our final graph will be given to us here with the purple graph being our cost function. Our profit function will be given here by the green graph. And we could change the scaling to make it easier to see.

Once again our welcome to new problem. This time we're dealing with specific costs. And remember that you could have fixed costs in a business. We could have fixed costs like, for example, rent. You could also have variable costs. Could have variable costs. Mm. Like, say, wages. Fixed costs weaken. Symbolize that of FC and variable costs V C, for example. And then off course, total costs. TC becomes the sum of fixed costs and variable costs. Fixed costs. They don't change month to month, but then variable costs do change based on, uh, the number of products being sold so you could have products costing a certain amount and, uh, being sold. On the other hand, we do also have things like, um, revenue. So we have revenue. So total revenue tr Okay is when you take when you take a specific number of items which you call Q. These are items, and you multiply that by the unit price on. That gives you the total revenue revenue being the amount of money that you get from sales. We also have marginal cost, which is simply the change in total cost off the changing number of units. So This is a change in total cost. Bus is change in, uh, number of units number of units produced, for example, when we think about marginal cost, always saying is through the extra cost that businesses have to deal with in producing on extra unit in producing an extra units. So that's the extra cost that we do have. Profit is obviously gonna be the difference between total revenue and total cost. That's the total profit T standing for total. We could also have marginal profit, uh, to optimize your profits. Your marginal cost has to equal to the marginal revenue. Remember, marginal revenue is changing our total revenue off a changing the quantity and the marginal cost is changing. Total cost of a changing, the quantities being sold. So in this particular problem, we have two questions. The first one is, uh, the find the prophet maximizing quality, find the profit maximizing quantity on. Then the, uh, the other one is visualized visualized the total total revenue, uh, together with total cost on the same graph on the same graph. And also, uh, visualize the marginal revenue. Imagine a revenue and marginal host roofs. So the first thing we're gonna do in this particular problem is to address, uh, the graphical issues, those of things we're looking at. So the first rough that we discuss is the total revenue. Uh, on total cost scruff. This is total revenue. Just remember that TR is total revenue, and TC is total cost rough. And, uh uh, that's based off the data provided so were provided with a quantity of units that that we're producing. And so we have to bring in specific data that reflects the wanted to Venice, for example. One unit is a variable cost of 20 and a fixed cost off 20 and then the total cost is on the floaty on. Besides that, we're not the total revenue, which is 20 and then the marginal revenue. It is also 20. So remember, we do have formulas for this. This is changing total revenue of a changing 20. And this is, uh, changing total cost of changing quantity. And then we have fixed cost total cost in bible cost. So taking these two together way get the total cost. Uh, that's why you have, for example, for quantity We want to say for pointing to one. For example, you get to see that the variable cost is 20. Fixed cost is 20. Total cost is variable cost plus fixed cost, which is 40 total revenue. Total revenue is how much money you get. Way actually do need after the total cost to me the marginal cost column. So let's change this a little bit. Mr. Movie. It's changing this a little bit. Okay, We're gonna be back truck. So this is variable cost? Yeah. On, then, Fixed cost on the in total cost. And then, of course, we have marginal cost. So fixed cost is 20. So if you add them, you get on. The total cost is 40 e. We're not gonna have any marginal costs because there's no change in, uh, total cost. Remember, if we do marginal cost, that's changing. Total cost off changing quantity. So at the beginning, you don't really have any change in total costs because you start with, um ah, specific quantity. So we have zero total revenues. 20 in marginal revenues 20. Selling two items. Fixed costs stays the same. So it's always gonna be 20. Fixed cost stays at 20. Hmm. So total cost will be 45 because you're summing up these two marginal cost is the change in total cost, and you can see if you want to change the total cost. In that sense, change in total cost would be 45 uh, minus 40. 45 is when you're selling two units and 40 is when you're selling one unit. So the change in total cost is five on, then divided by the change in quantity, you only changed by one unit. So two minus one, which is one. So this is 51 This is five. That's why we have marginal cost is five. When it comes to total revenue, we're doubling. And the reason why we're doubling is because the, um the that's that's gonna be the selling price. The selling price for the item, which is the revenue you get from Uh huh. Selling one unit. Total revenue for one unit is $20. If you want to think of it in terms of dollars, so he always selling to so at Q two, for example, if you go from the left to the right, you get to see that variable cost is 20 fixed cost. Sorry. Variable cost has given US 25 that's given fixed cost is 20 so total cost is variable cost plus fixed cost, which is 2025 plus 20 which is 45. We already have the marginal cost with computed right here, just five. And then we have the total revenue is the prize times the quantity we are surprise off or the selling price P is $20 so we have 20 times the quantity to which is 40. That's why right here we do have 40 as your total revenue and then your marginal revenue becomes the change in total revenue divided by the change in quantity uh, change in total revenue. You're seeing the revenue when two items are sold minus revenue. When one item was soul divided by tu minus one revenue and two items. Who sold this 40 when one item was sold, this 20/1. So this gives us 20 and we're just gonna go back and see that the marginal revenue becomes 20. So from this one will get to see that the marginal revenue is 20 and saw at each level off the way you're gonna proceed in the same light and fill up the table with the required data and each step of the way the computations reflect, uh, or they're subject to the formula is given. Yeah, 80 20. And then, of course, when you're variable, cost is 80 for four units or for five units, then the fixed calls stays the same. The total cost is the sum of variable and fixed. The marginal cost would be taking the total cost from five units minus the total cost from four units, which is 30 and then dividing by one. Because you're only changing a unit by one item, the total revenue you have five items each arch $20 so $100. And then finally, the marginal revenue you're taking these two different revenues 100 minus 80. Divide by one. So you get 20 right there seems like the marginal revenue never changes. That's the data we have. And, uh, looking at our graph, we can say while on the X axis we have quantity on then, on the Y axis, we have a total revenue or total cost. We get to see one item, two items, three items for items are items and then six items. Then on the on the while we have a zero, 20 40 60 80 100. And then 1 20. Yeah, total revenue. Total revenue you can see is this column Thistles The column for total revenue. And then if you profit of one in 20 for example, you have two and 40. Um so let's let's destroy envious, precise as possible. So you have one in 20 of two and 40 three and 60 your four and 80. You have five on a 100 and so on and so forth. So it's pretty much a straight line where if you only sell one unit, Yeah, you're gonna have 20 right there. So this is the total revenue growth on then, for the marginal revenue if you go through the process. Marginal revenue. Uh, rather, uh, that's the total revenue for the total cost. This one one at 40 to its 45 and then 55. We had 55 for 70. Um, Touro cost total costs. When you graph it, you're obviously going to start at 40 when it kind of crosses that on someone right here, then goes back and connects on 100 something similarly, and that becomes the total cost. So we do have to rough smooth. Top point is, this one is a total revenue, and then we have a total cost. The next issue that you have to deal with is the marginal revenue and marginal cost. Marginal. Well, earning was his marginal cost. Hmm. So in that sense mm. Right, you're gonna have. Well, we could start at one. And then we have 1.52 2.53 3.5 flowing. Mm hmm. Four point slides. I'm sorry. And we have 5.5. So this is on 05 10. Hmm. 15. 20. It's a process gruffly process. 25. Hmm. 30. Then we stop with 35. The marginal revenue, as you can recall, stays the same if you go back. You see, it's always gonna be 20. This is the marginal revenue right here. It always stays a Tony. So we go back and a graph straight line like that that reflects the marginal revenue. Staying at 20. But then the marginal cost, it rises up a little bit, for example, Uh, conceive that art one. It's That's pretty much zero. Then at two, it becomes five on then at three, we have turn, for example. But before we have, uh, 15 example at five, we have 25 for example. Something like that. A big fat. So the graph itself, which rises up a bit, turned it gets to three. Then he just bends up, promoted like that. So we just wanna make sure that 100% and, uh, so, based off that the yeah profit maximization profit maximization happens out about close to four units for mm hmm, a little bit beyond four units. So this is the point where profit maximization happens. I think we need to shrink this growth a little bit of that. So we're gonna bring it on something with blood. It's 100% exact, but yet the wound. So this is this is where marginal cost or marginal revenue equals two marginal cost. That's where you wanna optimize your profit. So it will say profit maximization happens, happens when marginal cost is the same. Was marginal revenue. Uh, but since you are producing whole units, profit maximization weapons, one q equals to float or unit. So once again, we had a, uh, economic problem while we were given a series Off items. Yeah, One item costing $20 to items 25 3 items study 54 55 80. And each one of the items had a prize off, $20 to be sold. And so we built a table showing fixed cost, which never changes. It's always $20 to produce, and then the total cost, which is the samples of variable and fixed cost total revenue is the product off the prize times the number of units castling, price times, number of units. And so if you produce one item, you're only gonna sell, get $20 you produce five items, you're gonna get $100. We use specific formulas to resolve for marginal cost and marginal revenue. Uh, went ahead and draft total revenue and total cost graph and then followed that. You can see this is kind of like a special location where profit is optimized because you have, uh, the maximum gap. So there is a maximum gap right here. That's usually where the profit is optimized, and mm, that's connected to this point on the marginal revenue marginal cost rough that uses for Ireland. So hope you enjoy the problem, feel free to send any questions or comments, and have a wonderful day

Okay, so we have our Sikh of X function, which is equal to 1000. He had a negative exports fine. And then r of X function, which is equal to 16 minus points. One x were asked to find the profit generated when telling by 50 tickets. So we have also intersection point here, so you can just integrate or find the area, which is a total profit from 2.9125 10 of our top graph, which is 16 minus points, one x minus wound 1000 each and little X when this file the X Okay, we could simplify this a bit. This turns into excellent, just 55 minus that. Okay, so let's take the integral of that. So that's just 55 X minus points one x squared older too. Wyness are actually plus 1000 needs it in negative effects evaluated at 5 50 and two points in one. Okay, so 55 times by 50 minus points one times 15 to park su divided. Right. So plus 1000 e to the negative 50. Okay, that gives me 151 to find. Minus 55. Evaluated at 2.91 minus puts one. What's 912 cartoon divided by two 1st 1000 I need you to negative two points. 91 Okay, that gives me 214.10 to 3 to 49. Okay. 15125 Minus this. Yes. Me. 14910 points. See 0.89768 I think I made a mistake here because I was supposed to evaluate from 0 to 5 15 so I have to break it up into two reports. Okay, so let's evaluate our remaining portion. So Mm. Is that correct? Let me stay in the problem. You know, it's fine. So this is our total profits.

Okay. The bus twitter visualized this is to graph it. Okay, As you can see, we appreciated area over here, and we have to intersection points over here. We have 2.906 calm of 59.7 and then we have over here. 550. Comment five. Which was given in the problem? Yeah, 500 tickets. Okay, so we have the interval me of the shaded area, and we know that our area is gonna be our backs minus cfx. So what's right out? An integral Our bottom bound is 2.906 And then we have 5500 on the top member or of X minus CIA backs minus segovax. This is equivalent to integrating 55 axeman 0.1 x squared over two plus 1000 e to the negative axe from 2.906 2 550 Okay. Plugging in the two bounds and subtracting should end up with 15125 minus 214 which is approximately 149 road 49,000. Um, 10 0.898 Because actually, there was a decimal on This is 214.102 Okay, So the total profit tickets is $14910.898.


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