This document discusses exponential growth and decay functions. It provides examples of how to set up and solve exponential equations to model population growth, fish population decline, world poultry production increase, and bacterial growth. It also explains how to calculate half-life and doubling time for exponential decay and growth. Examples are worked through to find the half-life of tritium decaying at 5.471% per year and a radioactive substance decaying at 11% per minute, and to determine the amount of radioactive substance remaining after 6 years given its half-life is 3 years. The document concludes with assignments for students.
Hierarchy of management that covers different levels of management
4.6 Exponential Growth and Decay
1. 4.6 Exponential Growth and
Decay
Chapter 4 Inverse, Exponential, and Logarithmic
Functions
2. Concepts and Objectives
Exponential Growth and Decay
Solve problems involving exponential growth and
decay.
Set up and solve exponential variation problems
using both common and natural logarithms.
Find half-life and doubling times.
3. Exponential Growth and Decay
The general equation of an exponential function is
y = abx
where a and b are constants.
Exponential growth occurs when b > 1.
Exponential decay occurs when 0 < b < 1.
The constant a is usually the starting value and b is the
percentage by which a is increasing or decreasing.
4. Exponential Growth and Decay
Example: In 2016, the population of a country was 70
million and growing at a rate of 1.9% per year.
Assuming the percentage growth rate remains constant,
express the population, P, of this country (in millions) as
a function of t, the number of years after 2016.
5. Exponential Growth and Decay
Example: In 2016, the population of a country was 70
million and growing at a rate of 1.9% per year.
Assuming the percentage growth rate remains constant,
express the population, P, of this country (in millions) as
a function of t, the number of years after 2016.
Starting quantity (a): 70
Growth rate (b): 1+0.019 = 1.019
Equation: P = 701.019t
6. Exponential Growth and Decay
Example: A population of fish (P) starts at 8000 fish in
the year 2015 and decreases by 5.8% per year (t). What
is the predicted fish population in 2020?
7. Exponential Growth and Decay
Example: A population of fish (P) starts at 8000 fish in
the year 2015 and decreases by 5.8% per year (t). What
is the predicted fish population in 2020?
Starting quantity: 8000
Decay rate: 1 – 0.058 = 0.942
Equation: P = 80000.942t
In 2020 (t = 5): P = 80000.9425
= 5934 fish
8. Exponential Growth and Decay
If the relationship is continuously growing or decaying,
the equation can be written
y = aekx,
where a and k are constants and e is the base of the
natural logarithm.
Exponential growth occurs when k > 0
Exponential decay occurs when k < 0
9. Exponential Growth and Decay
Example: World poultry production was 77.2 million
tons in the year 2014 and increasing at a continuous rate
of 1.6% per year (t). Estimate world poultry production
(P) in the year 2020.
10. Exponential Growth and Decay
Example: World poultry production was 77.2 million
tons in the year 2014 and increasing at a continuous rate
of 1.6% per year (t). Estimate world poultry production
(P) in the year 2020.
Starting quantity: 77.2
Growth rate: 0.016
Equation: P = 77.2e0.016t
After 6 years: P = 77.2e0.0166
84.98 million tons
11. Exponential Growth and Decay
How are you supposed to know which equation to use?!?
If you are given a formula, you don’t have to worry
about it—just use the formula they give you.
If the problem uses the word “continuous” or
“continuously”, use the equation with e.
If the rate isn’t given (and you are not solving for the
rate), it doesn’t matter which version you use.
If you’re still not sure, use the general form of the
exponential equation .
( )
x
y ab
12. Exponential Variation
Example: In a bacteria culture, there were 2000 bacteria
on Tuesday. On Thursday, the number has increased to
4500. Predict the number of bacteria that will be in the
culture next Tuesday.
13. Exponential Variation
Example: In a bacteria culture, there were 2000 bacteria
on Tuesday. On Thursday, the number has increased to
4500. Predict the number of bacteria that will be in the
culture next Tuesday.
Our starting quantity is 2000, and our starting time is
Tuesday. Thursday is 2 days away. This means that
2
2000 4500
b
2 4500
2000
b
2.25 1.5
b
14. Exponential Variation
Example (cont.):
Thus, our equation is
After 7 days, there should be
7
2000 1.5
C
34,172 colonies
2000 1.5t
C
15. Exponential Variation
Example (cont.):
If we had wanted to use the natural log equation (e), it
would have worked much the same way:
2
2000 4500
k
e
2 4500
2000
k
e
2
ln ln2.25
k
e
2 ln2.25
k
ln2.25
0.4055
2
k
0.4055 7
2000
C e
34,172 colonies
16. Half-Life and Doubling Time
Half-life refers to the length of time it takes for an
exponential decay to reach half of its starting quantity.
Doubling time refers to the length of time it takes for an
exponential growth to reach double its starting quantity.
Both of these problems are actually worked the same
way. Use the general equation unless the rate is
continuous; in that case, you would use the equation
with e. To find the half-life (or doubling time), let a = 1
and set the equation equal to ½ (or 2) and solve for t.
17. Half-Life and Doubling Time
Example: Find the half-life of
(a) tritium, which decays at a rate of 5.471% per year
(b) a radioactive substance which decays at a continuous
rate of 11% per minute.
18. Half-Life and Doubling Time
Example: Find the half-life of
(a) tritium, which decays at a rate of 5.471% per year
Since our decay rate is 0.05471, b will be 1–0.05471, or
0.94529.
1
0.94529
2
t
log0.94529 log0.5
t
log0.94529 log0.5
t
log0.5
log0.94529
t 12.32 years
19. Half-Life and Doubling Time
Example: Find the half-life of
(b) a radioactive substance which decays at a continuous
rate of 11% per minute.
“Continuous” means we must use the e equation.
Because this decays, our rate, 0.11, is negative.
0.11 1
2
t
e
0.11
ln ln0.5
t
e
0.11 ln0.5
t
ln0.5
0.11
t 6.30 minutes
20. Half-Life and Doubling Time
Example: If the half-life of 600 g of a radioactive
substance is 3 years, how much of the substance will be
present after 6 years?
21. Half-Life and Doubling Time
Example: If the half-life of 600 g of a radioactive
substance is 3 years, how much of the substance will be
present after 6 years?
Since the rate isn’t mentioned, it really doesn’t matter
which equation we use:
3 1
2
b
3 1
2
k
e
3
0.5
b
ln0.5
3
k
6
3
600 0.5 150 g
ln0.5
6
3
600 150 g
e