At the introductory stages of modeling, students should be expected to examine the mathematics behind a proposed model. For example, if they have reason to believe there are equal increases in y for equal increases in x, they should try a linear model. If they reason that increases are in a geometric progression, they should try an exponential model. This problem involves the development of a linear model at a basic level of mathematical sophistication and introduces students to the idea of curve fitting.
Problem. Shonda is interested in a job as a server at the Restaurante Ricardo to earn money while in college. She was told that servers are paid $12 per day for working the dinner shift on weekends plus tips. She was advised that servers averaged $4.50 per table served. Below is Shonda's analysis of her projected weekend (three-day) wages before taxes:
a. Extend Shonda's analysis by making the table show her wages when 7, 13, 35, and 40 tables are served.
b. Make a graphical presentation of the extended table from part a.
c. Are the wages increasing at a constant rate? (That is, are the data linear?)
d. Determine an algebraic model that expresses Shonda's projected wages as a function of the number of tables that she serves one weekend.
e. Graph the function on the same set of axes used in part b.
f. Estimate from the graph in part e. Shonda's projected wages for a weekend that she serves 40 tables. Now determine the answer algebraically.
g. How many tables must Shonda serve in order to project weekend wages of $200?
h. Do you think that it is possible for Shonda to earn $700 in one weekend? Carefully explain.
Notice that this problem involves using different scales on each axis. The scale on the vertical axis involves relatively large numbers. The writing assignment in part h gives students the opportunity to consider the range of function values for which the function has meaning.