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The price-demand and cost functions for the production of microwaves are given as 300 60 and C(c) 86000 + 30x where € is the number of microwaves that can be ...

Question

The price-demand and cost functions for the production of microwaves are given as 300 60 and C(c) 86000 + 30x where € is the number of microwaves that can be sold at a price of p dollars per unit and C(w) is the total cost (in dollars) of producing € units_a. Find the profit function in terms of €.P(c)Previewb. Evaluate the marginal profit function at € 1500 microwaves rounded to the nearest cent:P' (1500)per microwavePreview

The price-demand and cost functions for the production of microwaves are given as 300 60 and C(c) 86000 + 30x where € is the number of microwaves that can be sold at a price of p dollars per unit and C(w) is the total cost (in dollars) of producing € units_ a. Find the profit function in terms of €. P(c) Preview b. Evaluate the marginal profit function at € 1500 microwaves rounded to the nearest cent: P' (1500) per microwave Preview



Answers

Marginal cost A division of Ditton Industries manufactures the Futura model microwave oven. The daily cost (in dollars) of producing these microwave ovens is $$ C(x)=0.0002 x^{3}-0.06 x^{2}+120 x+5000 $$ where $x$ stands for the number of units produced. a. What is the actual cost incurred in manufacturing the 101 st oven? The 201 st oven? The 301 st oven? b. What is the marginal cost when $x=100,200,$ and $300 ?$

Okay. Hello and welcome. We're looking at chapter two, Section five, problems 64. We're looking with a basic business problem. Um, so the demand equation for a microwave is 1 40 minus point is your point 0001 x. All right, the cost equation is 80 x plus 150,000. So this is how much it costs to produce on this is the kind of, Let's see, this is the price. This is kind of like the demand a little bit. All right, So, um, the total profit is given by p equals ar minus C, which is the same as X times P minus C. So we don't have an equation for our, So we can't really use that formula. Very well. Um, I guess we can infer that r equals x times p, which we can use. We have a formula for Pete. So, um, profit equals x times this minus this. And we're told we want a profit of nine million, so I'll just say equals nine million. All right, so once we have this set up, we kind of read our way through the problem and bounce some ideas around until we get the formula set up. Now, it's just a matter of the math. We just need to solve this equation. So, um, let's start with distributing. So this would be 1 40 x minus 0.1 X squared minus 80 x plus 150,000 s o equals will just say nine mil for now because I want to put this in order and combine these two middle terms here 1 40 minus 80 is 60 x crisis equals nine million. So the question is, is this possible? So see if there's any easy ways to simplify this Well, since we have such a small number here and then such big numbers here, uh, anyway, we try to simplify. If we divide by 1000 for example, on both sides, we're just gonna be making one part of it harder. Eso let's subtract nine million from both sides to get this equal to zero and then, ah, the good old quadratic formula might bail us out. So plus 60 x and so um, 150,000 minus nine million. I believe it's going to be negative. 850,000 Let me go check. Oh, no, that's not right. It's gonna be eight million. 850,000, right? Yeah. Let's just double check that. Yeah, there we go. Okay. Eso were good. That equals zero. So once we have a equaling zero, we can use our quadratic e formula with this X equals negative B plus R minus, the square root of B squared minus four times a. Don't forget my negative there times, see? All right. So that's all going to be over two times a Yeah. You're really gonna want a calculator for this one. All right. So, really, the main thing we want to check to see if this is possible is look at the B squared minus four A c. Maybe for the future word problems. I'll just focus. Go straight to here since we've kind of worked through to have explained it. So 60 squared is going to be 3600. And the question is, is that going to be bigger or smaller than this product here? Eso Let's go ahead and find that product. Um, let's see. So eso a negative times negative is a positive. So we are subtracting that would factor into the to your answer there. Um, if this was adding that it wouldn't matter which one's bigger, it would be possible. Um, so if we multiply these two together, um, it's just moving the decimal place, so it's very possible. So every time I move the decimal place to the right here, I move it to the left. Here. 1234 stubby. 1234 So it before times 8 85 is what it boils down to eso. We could do that real quick here. Oops. I put two lines. All right, 24 times. A 30 to 6. 34 32. So that is 35 40. So, um, it's just double check my math here with a calculator. All right, Let's just take a look to make sure we didn't make any other bad assumptions. So, walking through the problem, we were given the price here. I wrote that number down correctly. How we got the cost here. We were told that the total profit is x times that demand equation minus the cost equation. Alright, I found a found a mistake we could work with, so we're subtracting the entire cost equation, so we need to distribute that negative. That might change things. So if we work that mistake through, just flip one sign and it's going to make a difference here because instead of, um, this it's going to be nine million. Negative. Nine million minus 150,000. So that gives me negative one million. 10 million. 50,000. Let me get myself straight here. Yes. Okay. Got my place values all mixed up. So it's not that drastic. Okay, there we go. So negative. 9,150,000. So that's going to change us down here. Still gonna multiply with this here. So 60 squared is still 30. 600. Um, we're still going toe. Use this trick here. 1234 123 four. So it's going to be 9, 15 times for this time. I'm glad I looked back over this six. 36. 60. Okay, so 3600 minus 36 60 is going to be a negative number. So you're gonna have a negative in a square root eso You can tell your boss at the microwave factory that it is not possible to make nine million with the current assumptions. Right? So our final answer there officially is no, not possible. All right,

They have the cost function, right, And it's expressed in terms of corny, and we need to find the marginal costs. Alright, marginal cost. And that is just a derivative of the cost. Right? So we just have to find a derivative here, So that is gonna be, you know. Ah, the derivative off that function is 3.24 Q squared because 75. Right, So that is gonna be the marginal cost function. So this is a martial cost function, and then the be part you need to find C 50 and see Prime of 50. So this is the A part, right? So the B parts C 50 reason this one gonna put 50 everywhere received que Right. So this is gonna be ah, just a cognitive don't work. So the cost is measured in daughters is gonna be that and then see prime of 50 see a prime of 50. We're gonna use the marginal function, this one that I'm cleaning here, and he used that one. So we're using this function. Okay. S o, What is happening is, uh, 0.24 So that is 6 75 donors, right? So, uh, this one is the 1st 1 here. This one, right? Is the the cost of producing, uh, 50 quantities. Right? So So that is, uh, the amount of dollars needed to produce, uh, 50 items. Right? So the cost of producing 50 quantities that they said and then a 2nd 1 Right? 2nd 1 This one, it is a marginal cost per item. Right. So, uh, is a marginal cost per item at a production point off 50 items, right? So? So that is they definition?

So for this problem, it's a little bit tricky without having to copy the graphs and everything here. But basically, for the first part we're looking for here is What's the cost of manufacturing 60 units of goods? Well, So then you had to look for where 60 is out here and it's what's going by tents down below here, it's going by hundreds Over here. And if you line up 60 with that, what you'll get here for silver party Should be 1100 for cf 60. Okay, So that should be the first part there. So for part B being asked, what's the marginal cost When 40 goods are manufactured? So you're looking for C prime of 40, which means you're going to use the second graph here. And when you look at 40, so 40 or so closer to zero, you line that up here and cooperate. So now for the See prime of X. The White Police go up by 2.5 here. Okay. And so how that relates to is so for that one that's going to be 12.5. Okay. So many dollars per unit there now for part C. So then it's asking when cfx is equal to 1200. Then you just have to look at the graph And like a line where 1200 is and that happens at x equals hundreds. That's for part C. There suffering T. So here is asking when to see prime of X Equal 22.50. So this one you're taking the second graf for C. Double percy prime and looking at where 22.5 is here. So that's going to be between 20 and Sort of like one line above 20 basically. And you have to markers for that that where that happens on the graph. 1st 1 is at x equals 20 second one's at X equals 1 40 sense for party. Okay, Last part. So for party saying where is the lowest value of C. Prime of X. So they're looking for the men of this? And so if you take a look, there's like a part of C. Prime, right? It it has a minimum. If you look below that, there's also see double prime crossing through A zero at the same time that this is occurring here. And so then that's Occurring at x equals 80. That's the first part of that. And then see prime of 80. Check that out as equivalent to $5 per unit.

So if we're given the cost, function C is equal to two thousand two plus thirty five hundred, where Q. Is the demand function The square root? Oh, fifteen thousand minus one point five p. And he is our price. We want to rewrite. You're right, all of the following terms with respect to our demand keep. So you might recall that revenue is how many items are being sold times the cost of it. So we know that the number sold is are the man. So this here will be Q. And then so actually, liberation. It called the cost because I already have a function called cost. But it should really be the how much each one is going for. It's actually just say what that is. That's really just the price that they're being sold for. Okay, so this is going to be Q. Times P. So since I want to write this in terms of Q, so I have to Caroline, he is in terms of so up here. We're told that the man function depends on P. So we could just use this to solve for what Pete iss. So to get rid of this wear room here will square each side so we'LL have Q so you need to square each side and then that'LL leave were fifteen hundred from Started fifteen thousand minus one point I'LL Go ahead and subtract fifteen hundred over It all gets Q Square Little Dividing line here, so Q Square minus fifteen hundred fifteen thousand is equal to negative one point five p and then to solve for PM Want to divide by negative one point five Get he is equal to you. Squared minus. So when the actual title so Q squared will be over negative one point Bye Negative one point and then this will be added to so fifteen thousand divided by one point five gives ten thousand And then I can rewrite this here as a fraction negative one over negative one point five to give negative two thirds you squared plus ten thousand. So now, although ahead and take this and plug it in for Pete So I'll get you times negative two thirds you square wass and some how this became one thousand sort of ten thousand. So plus ten thousand go ahead and distribute the cue and I get it negative two thirds you razed to the third power plus ten thousand times Q. And so this here will be our revenue function are. And since it's depending on cue local, are you now to find the prophet function depending on you? So first, remember, profit is equal to so it is our cost, not our costs. First, our revenue function so will be revenue. And then I'm going to subtract off our cost function. So now I just need to poke everything in negative two birds, you cute plus and thousand u And then this will be minus two thousand. You plus thirty, five hundred and simple Fine. It's down a little bit more. Get negative. Two herds feud, then ten thousand Q minus two thousand Hugh is eight thousand. You and then there is no constant and our revenue. So I'll just negative thirty five hundred. And so this here will be my profit function. So Petey or capital P based off, will you, then marginal. So remember, when we see the word marginal, it means a small change. So this is saying what is a very small change of our profit function And remember, in terms of what we know this means take the derivative So I'll have p prime of new So this will be equal to so Oh, and just change one thing so they should actually be positive thousand you since the ten thousand queue here is bigger than the minus two thousand Cute But I have here once I distribute the negative side So taking the derivative of this polynomial here that we have for the prophet will just be a use of our rule So we'll end up with negative too Q squared and then plus just eight thousand And this here will be our marginal profit. And then if I want to find the marginal profit when prices five thousand dollars, so are marginal profit function is depending on cue as opposed to pee So the first thing I need to do is find out what I'm right here. So p equals by thousand. Then what is it? You Thank you is equal to what? Well, earlier we said that we I could use the demand function to find out what Hugh is. So all I need to do is plug in five thousand into the demand function, so b square broom uh so fifteen thousand minus one point five and he is five thousand. So we end up with so fifteen thousand minus. So one point five times five thousand is seventy five hundred. And this here is the square root of seventeen five hundred When I subtract that you So you may simplify this a little bit if you want, but instead if we plug this value directly end to our marginal profit. So if I have p prime of the square roots square root of seventy five hundred, we'LL end up with so negative too square root of seventy five hundred Where'd both Hey thousand So square root of seventy five hundred squared is seventy, five hundred times negative too We'LL give negative fifteen thousand and then adding eight thousand to that Linus eight thousand start adding a thousand to it That will give us negative seven thousand. So the marginal profit wind we're trying to sell at five thousand dollars is negative at seven thousand dollars


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