M331 Introduction to Mathematical Modeling

Problem Set 1: Modeling with Functions

The write-up of problems 2, 5 and 6 should be typed, e.g., using LaTeX (free) or MSword. Due Friday September 8, 2000.

1. Consider the model for a baseball franchise where the marginal revenue is given by
   $$MR(q) = \beta$$
   and the marginal cost is given by
   $$MC(q) = 2q^2-20q+100$$

   a)

   If the management insists on hiring 10 star players, what is the minimum value of $\beta$ such that the team is profitable?

   b)

   If $\beta = 60$ what is the minimum number of stars required to break even?

   c)

   If $\beta = 60$ for $q \le 10$ and $\beta = -60$ for $q >10$ plot and interpret the model for the total revenue.

   d)

   Is it true that in general the maximum profit occurs when the value of the marginal cost is a minimum? Justify your answer.

2. Plot the graphs shown in Figure 1.21 for the franchise model in Problem 1. Interpret each graph as fully as you can. This problem requires a careful analysis to avoid omitting an important point. ADDITION: Let $\beta = 60$ in this problem.
3. Section 1.2, Problem 4. Hint: Start with the formula $MR(q+1) = TR(q+1)-TR(q)$.
4. Section 1.3, Problem 1. Hint: Consider the relative steepness of the supply and demand curves in this problem. Plot four graphs by selecting steep and flat combinations of the suppy and demand curve. Compare and draw conclusions.
5. Section 1.3 Problem 2. Be as succinct as possible.
6. Section 1.3, Problem 5. Itemize your factors.