Finding the Minimum Average CostDate: 2/12/96 at 12:27:1 From: Anonymous Subject: average cost / calculus I've had problems with this one. "The cost of producing x units of a certain product is given by C=10,000 + 5x + (1/9)x^2. Find the value of x that gives the minimum average cost." The problem also listed these following multiple choice answers: a) 30,000 b) 300 c) 3,000 d) 30 e) None of these Now, we have the correct answer, what we need is the actual way to do this problem. Thank You, Love, Erin Date: 2/13/96 at 18:35:11 From: Doctor Ethan Subject: Re: average cost / calculus Hi there. I think I have the method you're looking for. You have a cost function for x units given by: C = 10,000 + 5x + (1/9)x^2 Now, the cost per unit is going to be that same function divided by x. Let's call this function U: U = 10,000/x + 5 + (1/9)x To find the minimum total cost, you need to find the minimum of this function. Analyzing the derivatives should get you the answer you need: U' = -10,000/(x^2) + 1/9 U'' = 20,000/(x^3) (You might want to try this yourself before reading on...) By examining U'= 0, we find that x = 300 is a critical point. Further, since U'' is positive, the function must be at a minimum at that point. I sure hope I came up with the answer you were looking for. -Doctor Ethan, The Math Forum |
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