Crossroads in Mathematics

Example 7

Population Growth Comparison (Exponential Growth)

Students should have the opportunity to solve problems that lend themselves to the use of multiple strategies.

Problem. The population of Huntsville, Texas, was 25,854 in 1985 and is increasing by 1.55% each year. The population of Conroe, Texas, was 22,314 in 1985 and is increasing by 4.35% each year.

a. Make a table showing the changing population of the two cities.

b. In which year did or will Conroe's population first exceed Huntsville's?

Note: According to their respective Chambers of Commerce, the population of Huntsville was 23,936 in 1980 and 27,925 in 1990 and the population of Conroe was 18,034 in 1980 and 27,610 in 1990. This implies average annual growth rates of 1.55% and 4.35%, respectively, which can be used for forward or backward population projections from the given years. The 1985 populations in the stated problem are the geometric means of the census figures. This problem can be readily adapted to your locality.

The problem can be solved using (1) arithmetic, by building a table one line at a time; (2) recursion, on a calculator (see Figure 1); (3) a sequence defined by a recursive formula; (4) a sequence with an explicit nonrecursive formula; and by functions (5) numerically using tables (see Figure 2), (6) graphically using the ZOOM-IN feature of a graphing utility, (7) using the SOLVE feature of a calculator or computer software, or (8) the traditional way using logarithms.

Figure 1. Recursion on a large-screen calculator.

Figure 2. A table of values generated by algebraic formulas.