Science:Math Exam Resources/Courses/MATH104/December 2010/Question 04
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Question 04 |
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It costs a small firm C(q) dollars to produce q kilograms of a certain chemical, where The average cost of production per kilogram is defined to be Use calculus to find how many kilograms the firm should produce in order to minimize the average cost of production per kilogram. Please simplify. You need not justify that your calculation actually minimizes average cost. |
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? |
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! |
Hint |
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If you are looking for maximum or minimum values of a function, you can start by looking for its critical points. |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Solution |
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Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Let denote the average cost per kilogram of producing kilograms of this chemical, The domain of interest is because the company cannot produce negative kilograms of their product. In order to find the absolute minimum of the average cost function, we begin by looking for its critical points. The derivative of is This exists everywhere in our domain and is equal to zero when , that is, when Since , we can take the positive fourth root of both sides to obtain Hence, which is our final answer.
It is easy to check that this is indeed an absolute minimum by noting that when and when . Since is decreasing when and increasing when , must have an absolute minimum at Hence, the company should produce 1000 kg of chemical in order to minimize its average cost. |