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Firm knows that Its marginal cost for product I5MC 3x2 +250,its marginal revenue is MR = 300 2x, and its fixed cost is $125_What levcl of production will maximize p...

Question

Firm knows that Its marginal cost for product I5MC 3x2 +250,its marginal revenue is MR = 300 2x, and its fixed cost is $125_What levcl of production will maximize profit?Find the profit functlon:Ifthe Marginal Cost function Is glven byMC-Vx+and the cost of 100 items i5 S800, find the cost function

firm knows that Its marginal cost for product I5 MC 3x2 +250, its marginal revenue is MR = 300 2x, and its fixed cost is $125_ What levcl of production will maximize profit? Find the profit functlon: Ifthe Marginal Cost function Is glven by MC-Vx+ and the cost of 100 items i5 S800, find the cost function



Answers

If the cost of manufacturing $x$ items is $C(x)=x^{3}+20 x^{2}+90 x+15,$ find the marginal cost function and compare the marginal cost at $x=50$ with the actual cost of manufacturing the 50th item.

So, given the fact that we have or if we want to find the cost function given that the marginal cost is so see, Prime of X is equal to X plus one over X squared and that two units will cost five dollars and fifty cents. Well, so even if we weren't told that are marginal cost was the derivative of the cost function where we could do is recalled, that what marginal names is a very small change of something. So this is saying a very small change in the cost function and the way we look at very small changes in calculus is five. Oops, by the derivative. Okay, Rick. So and the unit cost being to put into two units costing by fifty will be So this names, if our cost function is C and CEO too, should be I dollars and fifty six. So if I want to find one possible see or one possible cost function, they can go ahead and intubate This first, our integrate our marginal called SOS heading. Right now that's C prime of X is equal to x plus so I can rewrite one over X squared as next to the negative second power Just because we are going to end up taking the interval of this for anti derivative. So I'll just go ahead and rewrite it like this first. So when we do take that tighter evidence it will be a little bit easier Sze easier for us to do so. So now I'm going to integrate side of this So I want you to go sign over a little bit like schools in a the X there. So I integrate side with respect to thanks And now the anti directive of the derivative or the marginal cause or just be our past function with them We can use the some product off the anti derivatives to relate this hour. So first, I'll just repeat what I have in there next to the ninety two and then distribute the anti revenue. So I went too far, I swear. So throw in anti derivative symbol cross and then we can apply the Powerball twice here. So remember exes really x to the first power, So this would be X plus one using the power room and then I define this by the new power too, and I have to add on some integration Constant call that C one Actually, since we already have a seat floating around for a variable really smart function, I'LL go ahead and call these d one instead. So this will be one plus So we'll be next to the negative too. Plus one and the divided bites Steam power which will be negative one and then plus our second integration Constant D too. And now, since I have to integration Constance, I could go ahead combined he's into one integration constant which Col. D Let's go, Harry. Like this little bit So again, one have squared minus thanks to the negative First power Plus de, there's one possible cfx and I'LL just go ahead and rewrite X to the negative First power as one over bucks. So he's heading Get with this power right here. So one over x. So I have this here now, So figure out what our constant d needs to be will go and use the fact that producing two units will cost five dollars and fifty cents. So I'm going to combine these two facts together and in doing so, I'll end up with so I know if I plug in to the cost function is to get by fifteen All right if I go ahead and plug to them. So first I'll just put some feces everywhere flying over the plug in and then plug to and for each of these places cut so I can go ahead and re right first five fifty as eleven over two and then so two squared is for divided by two will give me two and then I'll just be negative one half their plus de So we want to move everything over So this one's becoming eleven halves plus one How minus two is equal to D and then eleven house post one half will be twelve over to our six and then minus two will be for the end of the hall war. Now I'LL take my constant here that I found and the possible function And combine these two facts together to get that my cost function Given that if I produce too units I should get fifty We'll be one half ax squared minus one over x plus So this year will be our cost function

Cost function to presented by C dash and excess the number of units produced. So the marginal cost function is given. And we have to find a cost function that total cost function. So we have to integrate. See, we have to find out. See, for that we have to integrate C dash. So we integrate both sides and this is how it looks like. Uh, now, integration of X Cuba is using the power rule is X rays to the powerful before off X will be X squared over two. Plus, we have a constant off integral off. See as well. Now we're given another piece of information that the fixed costs are 6500. What is fixed cost means is, even if we don't produce any item is still there is some cost. So at X equal to zero, we have some cost, which is 6500 which will be able to see uh, eso if you place the see back here we get the final cost of production as total cost of production CXS X rays to the Paul 4/4, minus X squared over two plus 6000 500. So this is the final answer

Given the cost function, See of X. We want to know what is the marginal cost function. So to do that, we need to take the first derivative of our cost functions. See, Prime of X is three x squared plus two times 21 is 42 x plus, 110. That is our marginal cost function. We also want to know what is the marginal cost at X equals 100. So we're gonna need to find what is the value of our marginal cost function at one hundreds or substituting in 100. So that's going to be three times 100 I swear, plus 42 times 100 plus 110. That's going to calculate to be 34,000 300 and $10. That's the value of our marginal cost function. That X equals 100. We're also going to determine what is the actual cost to make that 100 item. So what we're gonna have to do is figure out how much does it cost to make 100 items? And how much does it cost to make 99 items and the difference between those two numbers will be. How much did it cost to make that 100 items? To do this, we go back to our original cost function, see of X, and we substitute 100 in for the axe. We have 100 cubed. I was 21 times 100 squared plus 110 times 100 plus 20. And that calculates to 1,221,000 and $20. If we substitute in 99 into original cost function, we have 99 huge plus 21 times 99 square plus 110 times 99 last 20. And that calculates to be 100 one million, a little ahead of ourselves there. 1,000,100 87,000 and $30. The difference between these two numbers is our actual cost to make the 100 the item and that number came out to be 33,000 $990. That's the cost. The actual cost to make the hundreds item

Given the cost function. C f X. We want to determine what is the marginal cost function. That means we need to take the first derivative of our cost function. So that's going to equal three X squared, plus two times 11 is 22 X plus 40. I also want to determine what is the marginal cost at X equals 100 items. That means we need to find see prime of 100. Well, just substitute that 100 right in so three times 100 squared plus 22 times 100 plus 40. And that calculates to be 32 1000 200 and 40. That's the marginal cost at 100 items. I would also like to know how much does it actually cost to make that hundreds item? To do that, we need to determine how much does it cost to make 100 items, and how much does it cost to make 99 items that the difference between those two numbers is going to be the cost of making the 100 item. So we're going to go back to our original cost function and substitute in 100 will get 100 huge plus 11 times 100 squared plus 40 times 100 plus 10 that is equal to one million, 100 and 14,000 and $10. We'll do the same thing to find the cost of making 99 items will substitute 99 into our original cost function. Times 99 squared less 40 times 99 plus 10 and that calculates to one million and 82,000 $80. When we take the difference between these two numbers, we get 31,000 900 and 30. That's the actual cost to manufacture the hundreds item.


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