Laura Martinez is interested in buying and selling soybeans on the Chicago Board of Trade. She remembers that they discussed concepts like price functions, revenue functions, cost functions, profit functions, marginal cost, marginal profit, marginal revenue, etc. in her college calculus class. She would like to use some of these concepts to help her make good business decisions. Her best guess is that the price function "p", in dollars, for selling "q" bushels of soybeans is . p(q)=10-(q/800). She has a supplier that will sell her the soybeans for $5 per bushel. She knows that she will have fixed expenses (handling fees, commissions, taxes, etc.). 

She estimates that her fixed costs will be $1200. You should prepare two reports: Report 1: A letter to Laura Write a letter to Laura giving her the results that are of interest to her. What she really wants to know is the result: given some price to be determined in the future, how does she decide how much to buy and what will be her expected profits. You can present the results in the form of a formula or a graph or both. She does not want to read any mathematics, but she wants to know that your results are based on mathematics. Give some idea of what you did, but explain your procedure and results in plain English. Imagine that a student majoring in economics will grade your letter to Laura. You may assume that the reader knows the meanings of the technical terms from economics. Report 2: A Technical Report: Write a report explaining the work that you did and what you learned. 

Assume that the reader is an "A" Calculus student at some other school. You do not need to explain minor details of the calculations. However, your reader does not know the meanings of the economic terms, so these must be explained. You must also explain your general procedures. Your report should be completely independent of the project instructions. It should have an introduction that briefly states the purpose of the work and the key results, followed by a body that is organized into sections for easy readability. Keep in mind that you are writing about the problem, not your experience of solving it. ("The result is." rather than "First I looked at." and "I found that the result was."). Assume that your technical report will be mailed to the reader with no additional documentation. Your reader will study the report and then take a quiz written by your professor. This means that the mathematics must be correct and well explained and the report must be readable and easy to study.

For a complete analysis of the problem you will need to include the following:

1. A revenue function, R(q), for selling q bushels at a price p(q)

2. A total cost function, C(q), for buying q bushels

3. A total profit function, P(q), for buying and selling q bushels

4. A graph of the three functions on the same coordinate system. (label your axes) What price per bushel should she sell the beans to maximize her total profit? How many bushels would she need to sell? What price per bushel?

5. A graph of the derivative functions R' (q), C' (q), and P'(q) on the same coordinate system. (label your axes). Do the graphs of R' (q) and C' (q) intersect? If so, describe the significance of the point of intersection. Does the graph of P' (q) intersect the q axis? If so, describe the significance of the point of intersection.

6. Compute the marginal profit for q = 2400. Describe what this means.

7. If the supplier raises the price of beans to $ 5.50 per bushel how many bushels should she buy to maximize the total profit?

8. The supplier changes the price so often that Laura decides to find a formula that will tell her what quantity, q, to buy at k dollars per bushel. Find this formula that will maximize her total profits.