CALCULUS CONCEPTS: Projects: Chapter 5

5
Analyzing Change: Extrema and Points of Inflection

Project 5.1: Hunting License Fees

Setting

In 1986, the state of California was trying to make a decision about raising the fee for a deer hunting license. Five hundred hunters were asked how much they would be willing to pay in excess of the current fee to hunt deer. The percentage of hunters to agree to a fee increase of $x is given by the logistic model ³⁹

Suppose that in 1986 the license fee was $100, and 75,000 licenses were sold. Suppose that you are part of the 1986 Natural Resources Team presenting a proposed increase in the hunting license fee to the head of the California Department of Natural Resources.

Tasks

1. Illustrate how the model can be used by answering the following questions.

What was the hunting license revenue in 1986?

Suppose that in 1987 the fee increased to $150.

a. What percentage of the 1986 hunters would buy another license?

b. How many hunters is that?

c. What would be the 1987 revenue?

Repeat the preceding analysis if the fee were increased to $300.

2. If the fee increase for 1987 were x dollars, find formulas for the following:

a. the percentage of hunters willing to pay the new fee

b. the number of the 75,000 hunters willing to pay the new fee

c. the new fee

d. the 1987 revenue

3. Use your formulas to determine the optimal license fee. Also, find the optimal fee increase, the number of hunters who will buy licenses at the new fee, and the optimal revenue.

Reporting

Prepare a letter to the Secretary of Natural Resources. Your letter should address the fee increase and expectations for revenue. You should not make it technical but should give some support to back up your conclusions.
Prepare a technical written report outlining your findings as well as the mathematical methods you used to arrive at your conclusions.

³⁹ Based on information in Journal of Environmental Economics and Management, vol. 24, no. 1 (January 1993).

Project 5.2: Fund-Raising Campaign

Setting

In order to raise funds, the mathematics department in your college or university is planning to sell T-shirts before next year's football game against the school's biggest rival. Your team has volunteered to conduct the fund raiser. Because several other student groups have also volunteered to head this project, your team is to present its proposal for the fund drive, as well as your predictions about its outcome, to a panel of mathematics faculty.

On the basis of your findings, predict the optimal selling price, the number of T-shirts you intend to print, the costs involved, the number of T-shirts you expect to sell before realizing a profit (that is, the break-even point), and the expected profit.

Task A

Follow the tasks for Project 2.2 on pages 141-142. You will find a partial price listing for the T-shirt company in Table 2.73 on page 142.

Task B

1. Getting Started: Review your work for Task A. If you wish to make any changes in your marketing scheme, you should do so now. If you decide to make any changes, make sure that the polling that was done is still applicable (for example, you will not be able to change your target market.) Change (if necessary) any models from Task A to reflect any changes in your marketing scheme.

2. Optimizing: Use the models of demand, revenue, total cost, and profit developed in Task A to proceed with this section.

Determine the selling price that generates maximum revenue. What is maximum revenue? Is the selling price that generates maximum revenue the same as the price that generates maximum profit? What is maximum profit? Which should you consider (maximum revenue or maximum profit) in order to get the best picture of the effectiveness of the drive? Re-evaluate the number of shirts you may wish to sell. Will this affect the cost you determined above? If so, change your revenue, total cost, and profit functions to reflect this adjustment and re-analyze optimal values. Show and explain the mathematics that underlies your reasoning.

Discuss the sensitivity of the demand function to changes in price (check rates of change for $14, $10, and $6). Does the demand function have an inflection point? If so, find it. Find the rate of change of demand at this point, and interpret its meaning and impact in this context. How would the sensitivity of the demand curve affect your decisions about raising or lowering your selling price?

Reporting

1. Write a report for the mathematics department concerning your proposed campaign. They will be interested in the business interpretation as well as in an accurate description of the mathematics involved. Make sure that you include graphical as well as mathematical representations of your demand, revenue, cost, and profit functions. (Include graphs of any functions and derivatives that you use. Include your calculations and your survey as appendices.) Do not forget to cover Task A in this report as well.

2. Make your proposal and present your findings to a panel of mathematics professors in a 15-minute presentation. Your presentation should be restricted to the business interpretation, and you should use overhead transparencies of graphs and equations of all models and derivatives as well as any other visual aids that you consider appropriate.