8 Repetitive Change: Cycles and Trigonometry

 Project 8.1: Seasonal Sales Setting Some companies sell products or offer services that are seasonal in nature. Summer or winter sports equipment is one example of a seasonal product, and swimming pool cleaning is an example of a seasonal service. Sometimes companies specialize in two different products or services for which the demand peaks in different seasons. For example, a snow ski shop may sell patio furniture in the summer. Tasks Find a company that specializes in a seasonal product or service and that is willing to release sales data. Obtain data from the company. The data should span at least 1 year. If the company specializes in two different seasonal products, you may need to consider sales for the two products separately. 1. Find a sine model for the data. Do you believe a sine model is the most appropriate model for the data you found? Give the period, amplitude, and vertical shift of the sine model. Interpret these values in the context of the company's sales. 2. Find the maximum and minimum sales for the model. When do these sales occur? Compare the extreme points of the model with the highest and lowest data values and corresponding times. According to the model, when are sales increasing and decreasing most rapidly? 3. If a sine model is not the best model for the data, find a more appropriate model. Explain your reasons for choosing a different model. Reporting 1. Write a letter to the company from which you obtained data. If the company sells only one product or offers only one service, suggest another product or service to offset the minimum sales created by the current product or service. Suggest when the company should begin advertising each product or service. If the company already sells two products or offers two services, analyze whether the peak sales of one offset the low sales of the other. If you believe the company might benefit from offering a third product or service or from increasing advertising at particular times during the year, offer suggestions in your letter. 2. Write a report for your files containing all the mathematics used in performing tasks 1 through 5 to use as reference should the sales manager for the company call to ask you questions.

Project 8.2: Lake Tahoe Levels

Setting

Lake Tahoe lies on the California&endash;Nevada border. Its level is regulated by a 17-gate concrete dam at the lake's outlet. By federal court degree, the lake level must never be higher than 6229.1 feet above sea level. The lake level is monitored every midnight. The United States Department of the Interior reports (8/2/96 ADAPS report) that the lake level on the first day of each month from October 1995 through September 1996 was as shown in Table 8.26.

TABLE 8.26

 Month Lake level (feet above 6220 feet above sea level) October 1995 6.37 November 5.93 December 5.68 January 1996 6.05 February 6.68 March 7.58 April 8.00 May 8.29 June 8.87 July 9.03 August (est.) 8.90 September (est.) 8.20

8. Discuss any other types of models that could be used to model these data. Fit any such models to the data. Discuss how well they fit the data, and use them to perform Tasks 6 and 7.

1. Examine a scatter plot of this set of data. Discuss why a sine model might be appropriate in this case. If a sine model were used to describe these data, estimate its period, amplitude, and vertical shift. Fit a sine model to the data. What are the model's period, amplitude, and vertical shift? Compare these answers to your expectations, and discuss any discrepancies. Rewrite your model so that its output will be in feet above sea level.

2. Carefully sketch a graph of the model for the lake level. Use the sketch to estimate the lake level in January 1996 and the rate at which the level of the lake was changing at that time.

3. Use the model to estimate the lake level in January 1996 and to estimate numerically how quickly the level of the lake was changing at that time.

4. Use the derivative of the model to find the rate at which the level of the lake was changing in January 1996.

5. According to the data, when was the lake at its lowest level? When was it at its highest level? On a sketch of the function for lake level, draw lines that are tangent to the graph of the model at the model's minimum and maximum points. What is important about the tangent lines at these points?

6. Use the derivative of the model to estimate the month and day between October 1, 1995 and September 1, 1996 when the lake was at its lowest level. Also determine when the lake was at its highest level. According to the model, did the lake remain below the federally mandated level between October 1, 1995 and September 1, 1996?

7. When (between October 1, 1995 and September 1, 1996) was the level of the lake changing most rapidly? What was the level of the lake and how quickly was it changing at that time?

Reporting

1. Write a report addressed to the Department of the Interior with your findings on Lake Tahoe levels. Keep in mind that this should be a nonmathematical report of your conclusions. Use graphs in your report as appropriate. Include mathematical support for your conclusions as an appendix. Refer to your appendix in the body of your report as appropriate.
2. Prepare a 10-minute presentation of your conclusions to be given to the Department of the Interior. You should be prepared to discuss the mathematics if questioned, but keep in mind that your target audience is not expecting a math talk. Use overhead transparencies and/or other visual aids to enhance your presentation.
3. Prepare a poster of your conclusions. The poster should be self-explanatory, attractive, and easily readable from 3 feet away. Show enough mathematics on the poster to support your conclusions.