Cost of Manufacturing X Items (average cost minimization)

In summary: XZlLCBJIGRvIHVzZSBhIHNvbHZlciBvZiB0aGUgY29udGVudCwgYXMgYSBzdWJtaXNzaW9uIG9mIGZvcm1hdGluZyAxMDAgaXRlbXMgZm9yIG1hcmdpYWxsIGNvc3QgZnVuY3Rpb24sIGFuZCBtYXJnaW5hbCBjb3N0LiBJZiB0aGVuIGludGhvc2VkIHRoZSBhdmVyYWdlIGNvc3QgZ
  • #1
theclock54
14
0

Homework Statement


If the cost of manufacturing x items is:
C(x) = (x^3)+21(x^2)+110x+20



Homework 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."


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.

 
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  • #2
theclock54 said:

Homework Statement


If the cost of manufacturing x items is:
C(x) = (x^3)+21(x^2)+110x+20



Homework 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."


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.

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
 

1. How is the average cost of manufacturing X items calculated?

The average cost of manufacturing X items is calculated by dividing the total cost of producing X items by the number of items produced. This gives an average cost per item, which can then be used to determine the most cost-effective production quantity.

2. What factors contribute to the cost of manufacturing X items?

The cost of manufacturing X items can be influenced by a variety of factors, including the cost of raw materials, labor, overhead expenses, and any fixed costs associated with production. Additionally, external factors such as market demand and competition can also impact the cost of manufacturing.

3. How can the average cost of manufacturing X items be minimized?

The average cost of manufacturing X items can be minimized by identifying and reducing inefficiencies in the production process, negotiating better prices for materials and labor, and maximizing economies of scale by producing larger quantities. Implementing cost-saving measures such as automation and streamlining operations can also help to minimize average costs.

4. What is the relationship between production quantity and average cost?

The relationship between production quantity and average cost is known as the economies of scale. This concept states that as production quantity increases, the average cost per item decreases. This is because fixed costs can be spread out over a larger number of items, resulting in a lower average cost per item.

5. How can the cost of manufacturing X items be forecasted?

The cost of manufacturing X items can be forecasted by analyzing historical production data, market trends, and any potential changes in costs of raw materials or labor. Additionally, conducting regular cost-benefit analyses and staying updated on industry developments can help in accurately forecasting the cost of manufacturing X items.

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