MATH104 December 2010
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q3 • Q4 • Q5 • Q6 •
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!
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.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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
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.
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Optimization, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag