Project 10.1: Snow Cover

Setting

Often contour maps of a geographical region are not topographical maps showing land elevation. Contour maps could indicate other features such as depth of the water table, richness of mineral deposits, population density, precipitation levels, or atmospheric conditions. You are probably most familiar with weather maps where contours called isotherms indicate temperature or contours called isobars indicate air pressure. The contour graph36 in Figure 10.34 shows the probability of snow cover in January.
36. W. Rudloff, World Climates (Stuttgart: Wissenschaftliche Verlagsgesellschaft, 1981).

FIGURE 10.34

(Click on the map to view it at full size: 1062 x 668 pixels)

1. Carefully examine the contour graph in Figure 10.34 below. You may wish to make several copies of the graph for use in this project. An enlarged copy may also be helpful. Begin by marking on a copy of the graph all the relative high and low percentages indicated in the graph. Use Hs and Ls to distinguish between high and low percentages.
2. Find a map or atlas of the world that indicates special topographical features. Choose at least five of the relative extreme points in Figure 10.34, and determine the precise topographical characteristic of the planet at each location that results in the corresponding relative extreme point.
3. What would you expect the snow cover probability on Mt. Everest to be in January? Locate Mt. Everest on the contour map. Does the graph indicate the probability you expected? If not, why not?

Reporting

Prepare a poster to present the contour map in such a way as to highlight the relative extrema and the corresponding topography at each point. Give some information about each point (such as latitude, longitude, elevation, and topography). Include on the poster the location of and information about Mt. Everest. Make the poster attractive and readable from a distance of 3 feet.

Project 10.2: Carbonated Beverage Packaging

Setting

A new firm that wants to compete with Pepsi and Coca-Cola has enlisted your services to design the optimal can shape for its product. The can is to hold 12 fluid ounces of a carbonated beverage.

Tasks

1. Find equations giving the volume and the surface area of a cylindrical can in terms of its height and radius. Use these equations to determine the height and diameter of a can that will hold 12 fluid ounces but will require the least amount of aluminum to construct.
2. Carefully measure the diameter and height of a 12-ounce can made by PepsiCo and of a 12-ounce can made by Coca-Cola. Compare these dimensions to the optimal dimensions you found in Task 1. Are there any beverage cans in the market that conform to the optimal dimensions?
3. Repeat Tasks 1 and 2, considering that it will cost twice as much to form the ends of the can as it does to form the cylindrical surface.

Reporting

1. Write a letter to the company directors explaining what dimensions they should use for their cans. Include your findings on the optimal dimensions as well as the dimensions of cans produced by competitors. Add your mathematical work as an appendix to support your conclusions.
2. Prepare an oral presentation to be given in front of the company directors. You should prepare visual aids in order to enhance your presentation. Your goal is to persuade the directors to construct cans according to the dimensions you suggest. Keep in mind that the company directors are probably not as concerned with all of the mathematical details as they are with your analysis of your findings. However, be prepared to answer mathematical questions in case any of the directors asks you to elaborate.