College Algebra 9th txtbk

340 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
5-2 Exponential Models
Z Mathematical Modeling
Z Data Analysis and Regression
Z A Comparison of Exponential Growth Phenomena
One of the best reasons for studying exponential functions is the fact that many things
that occur naturally in our world can be modeled accurately by these functions. In this section,
we will study a wide variety of applications, including growth of populations of
people, animals, and bacteria; radioactive decay; spread of epidemics; propagation of
rumors; light intensity; atmospheric pressure; and electric circuits. The regression techniques
we used in Chapter 1 to construct linear and quadratic models will be extended
to construct exponential models.
Z Mathematical Modeling
Populations tend to grow exponentially and at different rates. A convenient and easily understood
measure of growth rate is the doubling time—that is, the time it takes for a population
to double. Over short periods the doubling time growth model is often used to model
population growth:
A A 0 2 t d
where
A Population at time t
A 0 Population at time t 0
d Doubling time
Note that when t d,
A A 0 2 dd A 0 2
and the population is double the original, as it should be. We will use this model to solve
a population growth problem in Example 1.
EXAMPLE 1 Population Growth
According to a 2008 estimate, the population of Nicaragua was about 5.7 million, and that
population is growing due to a high birth rate and relatively low mortality rate. If the population
continues to grow at the current rate, it will double in 37 years. If the growth remains
steady, what will the population be in
(A) 15 years?
(B) 40 years?
Calculate answers to three significant digits.
SOLUTIONS
We can use the doubling time growth model, A A 0 (2) td with A 0 5.7 and d 37:
A 5.7(2) t37 See Figure 1.