Transcript Exponential Modeling

Exponential
Modeling
Section 3.2a
Let’s start with a whiteboard problem
today…
Determine a formula for the exponential function whose graph is
shown below.
x
f  x   f0 b
f  0  f0 b  3  f0  3
0
f  4  3 b  1.49
1.49
 0.839
b 4
3
4
(0,3)
(4, 1.49)
f  x   3 0.839
x
Constant Percentage Rates
If r is the constant percentage rate of change of a population,
then the population follows this pattern:
Time in years
Population
 P0  Initial population
3
P  0
P 1  P0  P0r  P0 1  r 
2
P  2  P 1  1  r   P0 1  r 
3
P  3  P  2  1  r   P0 1  r 
t
P  t   P0 1  r 
0
1
2
t
Exponential Population Model
If a population P is changing at a constant percentage
rate r each year, then
P  t   P0 1  r 
t
where P0 is the initial population, r is expressed as a
decimal, and t is time in years.
Exponential Population Model
P  t   P0 1  r 
t
If r > 0, then P( t ) is an exponential growth function, and its
growth factor is the base: (1 + r).
Growth Factor = 1 + Percentage Rate
If r < 0, then P( t ) is an exponential decay function, and its
decay factor is the base: (1 + r).
Decay Factor = 1 + Percentage Rate
Finding Growth and Decay Rates
Tell whether the population model is an exponential growth
function or exponential decay function, and find the constant
percentage rate of growth or decay.
1.
P  t   782, 248 1.0135
t
1 + r = 1.0135  r = 0.0135 > 0
 P is an exp. growth func. with a growth rate of 1.35%
2.
P  t   1, 203,368 0.9858
t
1 + r = 0.9858  r = –0.0142 < 0
 P is an exp. decay func. with a decay rate of 1.42%
Finding an Exponential Function
Determine the exponential function with initial value = 12,
increasing at a rate of 8% per year.
P0  12
r  8%  0.08
P  t   12 1  0.08 
t
 12 1.08
t
Modeling: Bacteria Growth
Suppose a culture of 100 bacteria is put into a petri dish and
the culture doubles every hour. Predict when the number of
bacteria will be 350,000.
First, create the model:
Total bacteria after 1 hour:
Total bacteria after 2 hours:
Total bacteria after 3 hours:
Total bacteria after t hours:
200  100  2
2
400  100  2
3
800  100  2
P  t   100  2
t
Modeling: Bacteria Growth
Suppose a culture of 100 bacteria is put into a petri dish and
the culture doubles every hour. Predict when the number of
bacteria will be 350,000.
Now, solve graphically to find where the population function
intersects y = 350,000:
t  11.77
Interpret:
The population of the bacteria will be
350,000 in about 11 hours and 46 minutes
Modeling: Radioactive Decay
When an element changes from a radioactive state to a
non-radioactive state, it loses atoms as a fixed fraction per
unit time  Exponential Decay!!!
This process is called radioactive decay.
The half-life of a substance is the time it
takes for half of a sample of the substance
to change state.
Modeling: Radioactive Decay
Suppose the half-life of a certain radioactive substance is 20
days and there are 5 grams present initially. Find the time
when there will be 1 gram of the substance remaining.
First, create the model:
Grams after 20 days:
Grams after 40 days:
Grams after t days:
 2
1.25  5  1 
2
2.5  5 1
20 20
40 20
 2
f t   5 1
t 20
Modeling: Radioactive Decay
Suppose the half-life of a certain radioactive substance is 20
days and there are 5 grams present initially. Find the time
when there will be 1 gram of the substance remaining.
Now, solve graphically to find where the function intersects
the line y = 1:
t  46.44
Interpret:
There will be 1 gram of the radioactive substance
left after approximately 46.44 days (46 days, 11 hrs)