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(7) A company estimates that the cost (in dollars) of producing x units of product can be modeled by C 900+0.04x 0.0002x",* >0 Find the production level tha...

Question

(7) A company estimates that the cost (in dollars) of producing x units of product can be modeled by C 900+0.04x 0.0002x",* >0 Find the production level that minimizes the average cost per unit.

(7) A company estimates that the cost (in dollars) of producing x units of product can be modeled by C 900+0.04x 0.0002x",* >0 Find the production level that minimizes the average cost per unit.



Answers

A company estimates that the cost in dollars of producing $x$ units of a product is given by $C=0.0001 x^{3}+0.02 x^{2}+0.4 x+800 .$ Find the production level that minimizes the average cost per unit.

This problem were using optimization to find the, um the minimal average cost per unit. So they said the Kanis per unit would be see over X ray. So and nobody give this some constant h justice. So it doesn't look as confusing. So in this case, were to divide this value by X to get a judge. So this simple but be 0.5 x plus 15 plus 504,000 rex. Now, um, it appreciate the derivative of this once, um, which we wouldn't need in such a zero. To find the optimization Asia Prime of X would be 0.5 plus X minus 5000 over X squared. So simplifying this will get that spot You. So, um, this is age prime tracks into the speaker to zero, and we try to find the minimal units. So ah, you know this because it can't be Well, it could be zero. Um, however, that's not really in this case. We we'll focus on this top function and, um, try to find the value of excellence gives us So this looks a little if you use that we're gonna use the contract formula negative people's air minus through peace group minus four A C over two A. And plugging in our values we find. But this is negative. One plus or minus one plus 1000. 10,000. Good. Over one. So, uh, this simplifying gets a value. Um, that is that roughly looks like negative, um, 100 over and plus 100. So because that's gonna complicated against provide this to having us having the minimal amount of sub units US 100.

For the given question. That total cost is given here now make you find the average cost will be see by exhorted here upon five X plus 15. That's 5000 my ex. Now point off Minimum cost D by D. X off see by X will be called to zero. So from this wicked differentiation off this function will be called to Europe on five minus 5000. Expert is it for 20 So from this record, X squared is equal to 5000 budget upon fight, so solving this figured exit for $200.

Alright In this problem, we're going to be minimized. The average function as defined to be like this zero point five x squared plus 10 x plus, uh, 7200 over X. So let's try to simplify this function. You get 0.5 X plus 10 plus 7200 over x. Okay, So if I would like to minimize this average function, I have to find a jury of the two of this expression and Saturday equal to zero to come up with the critical values. So if I take the derivative, we get 0.5. So let's say see, Prime music has to be called zero, and from there 0.5 and minus 772 100 over X squared has to be equal to zero. So let's try to solve this equation for X. So we get 0.5 X squared equals 7200. And if I divide each side by 0.5 and then we get X squared equals, uh 14, 144 and actually 14,400. And if you take the squared of both sides and then we get from there, we get X equals 120. Okay, so we got this critical value. But we have to test this critical value, which means that the minimum of this average function is occurring at this point. So that's why to be able to test that, we're going to be using the second derivative. So we're gonna be using this second derivative test. So now we need to come up with the second derivative and what is the second derivative? If I take the derivative of this expression again and we come up with 14,400 over execute and then we need to plugging this critical value into the second derivative. So we get, see double private 121 which is 14,400 over 121 120 cube. So this expression will be simplified into 100 actually one over 120 which is bigger than zero. So according to the second derivative test, if I plugging the critical value into the second relative, if the second derivative turns out to be positive that at that critical valley we have local A minimum. So that's why we have just proved that this average function is going to have minimum at X equals 120

Cochin is given that c equals 2 500 Plus 300 eggs minus 300 log X. Where the value of X should be greater than equals two. When we have to find the average cost function C. And analytically find the minimum average cost. So to solve discussion firstly we will write first part of the caution which is saying Bar C equals to be no. See upon X the value of C is given 500 upon X. Here 300 X upon X. So X. And X will be cancelled out minus 300 will be logged. It is X log X upon X. No it can be written in this 500 divided by x -300, taking as a common. So in the bracket -1 plus log eggs upon eggs. Now we can see this is the answer of the first part. No 2nd part of the question. Be back. We have to find bar C dash. It means we have to differentiate the situation. So when we differentiate it we get as the differentiation of went upon excess minus went upon excess pair Now -300 as common. So it will be after differentiation I get X into even by X minus law eggs you very little about X is Square. So it can better tennis -500 Upon excess with -300 X. And X will begins allowed. So in the bracket one minus log X. A born X is good. So we know according to the rule the setting setting The compliment bars should be equal to zero. It means we can died 500 equals 2 -300 In the bracket when minus logs it means 500 will weekends. Hello? So it can be a tennis 513 It was to Log X -1. So when I take minus went to the left side it will be five by three. Plus one equals two logs taking L C. M. And solving it. So it will be eight by three equals two logs. It means it means it can be done. Is X equals to edo the power X by three according to law route will be equals to 14.39 So bar see 14.39 will be equal to approximately 2 79.15. So we can say by the 1st derivative test post derivative is this is uh minimum, is a minimum. And also and also we can see for the graph that if we draw the graph for this value we can see the grass will be like a graph will be like this goes like this. So it means Farsi will be equal to 2 7. We can see when X is it was to 14.39 is the minimum. This is the final answer of the question. Who


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