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Tne ! marginal cost function for a manufacturer's product is 0003q} 0.034 + 4, where cis in dollars If fixed costs are $5000, determine the manufacturer'$...

Question

Tne ! marginal cost function for a manufacturer's product is 0003q} 0.034 + 4, where cis in dollars If fixed costs are $5000, determine the manufacturer'$ average cost function _0) O 0.OC01q' 0.01Sq? 4q S00u 6) @ 0.000lq 0.0154- 1 c) @ 0.0001q? 0.0154 0.O00lq? + 0.01Sq AM 0.O001q' _ 0.01S4? + 4q 2500 Bos birak

Tne ! marginal cost function for a manufacturer's product is 0003q} 0.034 + 4, where cis in dollars If fixed costs are $5000, determine the manufacturer'$ average cost function _ 0) O 0.OC01q' 0.01Sq? 4q S00u 6) @ 0.000lq 0.0154- 1 c) @ 0.0001q? 0.0154 0.O00lq? + 0.01Sq AM 0.O001q' _ 0.01S4? + 4q 2500 Bos birak



Answers

Business and Economics 45 - 52 cost Find the cost function for each marginal cost function
$$C^{\prime}(x)=0.05 e^{0.02 x},$ fixed cost is $\$ 5$$

Here. We're told that a sea of X is a cost of producing X units, then see of X over axes, the average cost per unit for part A. We want to show that if the average costs is minimum, then the marginal cost is equal to the average cost. So here we just have to set the derivative of the average cost equal to zero and then solve. So that is the average cost. So when we go to differentiate, we can just use quotient room, get that that IHS X time see prime of X minus sea of X all over X squared and then setting that equal to zero we get that X time. See, Prime of Axe is equal to see a vax. And that means that see, Prime of X is equal to see of ex all over axe. And that's what That's just what we wanted because, see, prime of axe, that's our marginal cost and then see of X over X. That's the average cost. So that's our solution for party. And then when we get to Part B, were given the cost function 16,000 plus 200 acts plus four times X to the three halfs and we want to find for part one we want to find the cost, the average cost and the marginal cost at 1000 units. We know that the cost was given to us and that's just see of 1000 and that is approximately 342,000 491. And so that's our cost. And then we want the average cost. So that's just see of 1000 all over 1000 and thats approximately 342 point for nine. And then next we want the marginal cost and the marginal cost is just see prime of 1000. So when we differentiate our cost function, we get 200 plus three halfs times, for which is six times the square root of 1000 and thats approximately equal to 389 0.74 And that was our marginal cost at 1000 units. So that is the solution for B. We'll call this be one and now the second part, ask us. It's asking us to find the production level that minimizes theatrics cost, So here we just do the same thing. We differentiate the average cost. But to do that, let's first start with finding the average costs. We know that that's just Segovax over axe and see if X was given to us. So that's 16,000 plus 200 acts plus four times X to the three halfs all over acts. And then we can break this up. We get that that 16,000 over X plus 200 plus four times X to the 1/2. And now when we take the derivative of our average costs, we get that that's minus 16,000 over X squared, plus two all over the square root of X. We want to set that equal to zero. So now we have that 16,000 all over. A tune is equal to X square and all over route acts, which means that 8000 is equal to X to the three halfs and then solving for X, we get that X is equal to 400. And that's the solution for part two of being, because we just wanted to find the production level that minimizes that average cost, and we know that that is a minimum for the following reason. So if we take the limit as X approaches zero of this function, we get that that's infinite, that's infinity, and then the limit as X approaches. Infinity of this function is also infinity. So that means that the value, the critical value that we found must be a minimum value. And so now the last part says to find the minimum average cost. So here we just have toe take CF 400 all over 400 because X equals 400 is where the minimum occurs. And if we do that, we get that that is equal to 320 and that completes the problem.

Affirms marginal cost function is three Q squared plus six Q plus nine. So we need to write a differential equation for the total cost. So the total cost gonna be deep sea of Q over D. Q. Is equal to the function they gave us. So three U. Squared six Q. Last nine. And then for part B find the total costs. If the fixed costs are $400. So now we integrate both sides. So if we take the integral of the left side we get cf Q. The inter role of three q squared is cute. three Q cubed over three Which is three cute. And then the integral of six Q. Is six Q squared over two. So that's three Q squared And then nine q. And then plus C. But they said the fixed costs are $400. So there is your plus C.

We have another marginal cost problem. This time we're going to be integrating, too. Integrate the marginal cost to get the total variable cost. So we're going to be using these integration features. One the integral of that. Some No, the integral of a Some equals some of the integral roles going to use the integral of X to the N d. X equals X to the n plus one over in plus one. When you use that several times and the definite undergrowth to be of a constant, he asks. Because B minus a plans to constant now our functions at hand. Are the world were asked for? Were asked for What is the total cost? What is the fixed cost and what is the total variable cost? And as we know, the total cost is equal to fixed costs plus total variable costs. So we'll be finding those out now are marginal. Cost function is which will designated C prime que is you squared minus 16 Q plus seven. So this is dollars per unit. The cost of making dollars per unit. One unit. It's the cost of making the next one after cute. When you integrate this from 0 to 20 because we want to know what is the total cost for making 20. The integral of that function is I think I'll put that up there from 0 to 20 is simply value at 20 minus the value zero the value of those terms of zero zero and so, but we'll put them there anyway. It's Cube Cube over three minus 16. Q squared over two plus 70. So the integral of those is 8000 over three minus 6400 over to plus 1400 which is equal to 866.67. This is the total variable cause of making 20. We know the fixed cost is 500 were given that originally because it's C zero, So the total cost is 500 plus 8. 66 0.67 equals 13 66 67. This is part sees final answer. He says. Part B's well, that's actually part days. Final nature. These are your final answers

So we're going to derive are marginal cost function from this cost function. C. So the marginal cost function M. It's just equal to the first derivative of our cost function or C. Prime. So we can say M is equal to the derivative 55,000 plus 470 times x minus 0.25 times X square. The first thing we're gonna want to do is split this derivative up since we have a derivative of one term plus another term minus a third term. So we can split that up into the derivative of the first term plus the derivative of the second term minus the derivative of the third term. And the derivative of a constant here are constant is 55,000 is equal to zero. So you can just get rid of that. And now we just need to deal with these two derivatives. And the thing that we're going to want to do is pull out each of these constants since it's just a constant times X. So we can bring that out of our derivative and multiply it by the derivative of just X and X squared. So this is equal to 470 multiplied by the derivative of x minus 0.25 multiplied by the derivative of X square. And to find the derivative of X and X squared. We just need to use the power rule. So here we have X to the first and here we have X squared. So the derivative of X is just one. So it's just gonna be 470 minus 0.25 and then derivative, the derivative of X squared is two times X. So 3.25 times two X. Which then simplifies to 470 minus 4700.5 X. Or I'm going to note at one half times X.


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