1. Applications and Models: Growth and Decay
Section 4.6
Applications and Models:
Growth and Decay

2. Population Growth
The function P (t ) = P0e kt, k > 0 can model many kinds of population growths.
In this function:
P0 = population at time 0,
P = population after time,
t = amount of time,
k = exponential growth rate.
The growth rate unit must be the same as the time unit.

3. Example Population Growth of the United States.
In 1990 the population in the United States was about 249 million and the exponential growth rate was 8% per decade. (Source: U.S. Census Bureau)
Find the exponential growth function.
What will the population be in 2020?
After how long will the population be double what it was in 1990?

4. Solution At t = 0 (1990), the population was about 249 million.
We substitute 249 for P0 and 0.08 for k to obtain the exponential growth function.
In 2020, 3 decades later, t = 3. To find the population in 2020 we substitute 3 for t:
The population will be approximately million in 2020.

5. Solution continued
We are looking for the doubling time T.

6. Interest Compound Continuously
The function P (t ) = P0e kt can be used to calculate interest that is compounded continuously.
In this function:
P0 = amount of money invested,
P = balance of the account,
t = years,
k = interest rate compounded continuously.

7. Example
Suppose that $2000 is deposited into an IRA at an interest rate k, and grows to $ after 12 years.
What is the interest rate?
Find the exponential growth function.
What will the balance be after the first 5 years?
How long did it take the $2000 to double?

8. Solution
At t = 0, P (0) = P0 = $2000.
Thus the exponential growth function is

9. Solution continued

10. Solution continued The exponential growth function is
The balance after 5 years is

11. Solution continued

12. Growth Rate and Doubling Time
The growth rate k and the doubling time
T are related by
kT = ln 2 or
* The relationship between k and T does not depend on P0.

13. Example A certain town’s population is doubling every 37.4 years.
What is the exponential growth rate?
Solution:

14. Models of Limited Growth
In previous examples, we have modeled population growth. However, in some populations, there can be factors that prevent a population from exceeding some limiting value.
One model of such growth is
which is called a logistic function.
This function increases toward a limiting value a
as t approaches infinity.
Thus, y = a is the horizontal asymptote of the graph.

15. Exponential Decay
Decay, or decline, of a population is represented by the function P (t ) = P0e kt, k > 0.
In this function:
P0 = initial amount of the substance,
P = amount of the substance left after time,
t = time,
k = decay rate.
The half-life is the amount of time it takes for half of an amount of substance to decay.

16. Graphs

17. Example
Carbon Dating. The radioactive element carbon-14 has a half-life of 5750 years. If a piece of charcoal that had lost 7.3% of its original amount of carbon, was discovered from an ancient campsite, how could the age of the charcoal be determined?
Solution: The function for carbon dating is
P (t ) = P0e t.
If the charcoal has lost 7.3% of its carbon-14 from its initial amount P0, then 92.7%P0 is the amount present.

18. Example continued
To find the age of the charcoal, we solve the equation for t :