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Cost of Manufacturing X Items (average cost minimization)

by theclock54
Tags: calculus, cost, function, math
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theclock54
#1
Nov16-11, 08:47 PM
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
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Ray Vickson
#2
Nov16-11, 09:13 PM
Sci Advisor
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P: 4,959
Quote Quote by theclock54 View Post
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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