Applications of
Exponential
Functions
Radioactive Decay
The amount A of radioactive material
present at time t is given by
t
h
0 initial amount at t=0
Where A0 is the
and h is the material’s half-life.
A A 2
Example 1:
The half-life of radium is
approximately 1600 years.
How much of a 1-gram
sample will remain after
1000 years?
Example 1
Solution:
A  A0 2
t
h
Given A0  1 , h  1600, and t  1000
1000
A  1 2
 0.648419777
1600
0.65 g remains
after 1000 yrs.
Oceanography
The intensity I of light (in lumens)
at a distance x meters below the
surface of a body of water
decreases exponentially by:
I  I0k
x
where I0 is the intensity of light
above the water.
Example 2:
For a certain area of the
Atlantic Ocean, I0=12 and k=0.6.
Find the intensity of light at a
depth of 5 meters in this
body of water.
Example 2:
Given I0=12 and k=0.6 and x=5 :
I  I0k
x
I  12  0.6 
5
I  0.93312
lumens
Malthusian
Population Growth

Malthusian model for Population
Growth assumes a constant birth
rate (b) and death rate (d). It is
kt
as follows: P  P e
0
where k=b - d , t is time in years,
P is current population, and P0
the initial population.
Example 3:
The population of the U.S. is
approximately 300 million people.
Assuming the annual birth rate is
19 per 1000 and the annual
death rate is 7 per 1000. What
does the Malthusian model
predict the population will be in
50 years?
Example 3:
Given: b=0.019 , d=0.007,
P0 =300 million, t=50
P  P0e
kt
where k  0.019  0.007  0.012
P  300, 000, 000 e
 0.012  50 
 300, 000, 000 e
 546, 635, 640 Prediction in 50 years
0.6
Epidemiology
Example 4:



In a city with a population of
1,200,000, there are currently
1,000 cases of infection with HIV.
Using the formula:
1, 200, 000
P
1  1, 200  1 e0.4t
How many people will be infected
in 3 years?
Example 4:
1, 200, 000
P
0.4 t
1  1, 200  1 e
Substitute 3 for t :
1, 200, 000
P
0.4 3
1  1, 200  1 e
 3,313
Infected in
3 years
Example 5:
Example 5:
Using our graphing
calculator, the
approximate
intersection of the
two functions at
(71,4160) gives us
the prediction:
In about 71 yrs
the food supply
will be outstripped
by population of
about 4160.