# Cost of Manufacturing X Items (average cost minimization)

by theclock54
Tags: calculus, cost, function, math
 P: 16 1. The problem statement, all variables and given/known data If the cost of manufacturing x items is: C(x) = (x^3)+21(x^2)+110x+20 2. Relevant equations All right, so the first few questions asked for total cost of producing 100 items, and marginal cost. I understood those well. Then it asked for the average cost function, which I found to be C(x)/x. I have a problem where it asks "The production level when the average cost is minimized." 3. The attempt at a solution Well, I would take the derivative of the average cost function (C(x)/x) which would give me (2(x^3)+21(x^2)-20)/(x^2) So then we set that equal to zero, I'm stuck at setting the top equal to zero. How can I factor it? I graphed the average cost function and found x to be approximately -10.40768. Is there any way to factor it? Or do I have to use a calculator? Thank you in advance for your replies. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
HW Helper
P: 4,159
 Quote by theclock54 1. The problem statement, all variables and given/known data If the cost of manufacturing x items is: C(x) = (x^3)+21(x^2)+110x+20 2. Relevant equations All right, so the first few questions asked for total cost of producing 100 items, and marginal cost. I understood those well. Then it asked for the average cost function, which I found to be C(x)/x. I have a problem where it asks "The production level when the average cost is minimized." 3. The attempt at a solution Well, I would take the derivative of the average cost function (C(x)/x) which would give me (2(x^3)+21(x^2)-20)/(x^2) So then we set that equal to zero, I'm stuck at setting the top equal to zero. How can I factor it? I graphed the average cost function and found x to be approximately -10.40768. Is there any way to factor it? Or do I have to use a calculator? Thank you in advance for your replies. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
You get a cubic equation having two negative roots and one positive root. There are formulas available for solving cubic equations, but they are rarely used. A numerical method is preferable in this case. Only positive roots make sense in this problem: we cannot have a negative production level!

RGV

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