Transcript Logistic Models - Northland Preparatory Academy

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Section 3.2b
In the last section, we did plenty of
analysis of logistic functions that
were given to us…
Now, we begin work on
finding our very own
logistic functions!!!
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Find the logistic function that has an initial value of 5, a
limit to growth of 45, and passing through (1, 9).
c
First, recall the general equation: f  x  
x
1 a b
Limit to growth = c  c  45
Initial Value = 5  Point (0, 5)
45  5  5a
45
45
f  0 
5
5
0
1 a b
1 a
a 8
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Find the logistic function that has an initial value of 5, a
limit to growth of 45, and passing through (1, 9).
Use the point (1, 9) to solve for b:
45
9
1  8b
45  9  72b
1
b
2
45
f 1 
9
1
1 8 b
Final Answer:
f  x 
45
 2
1 8 1
x
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Find the logistic function that has an initial value of 19, a
limit to growth of 76, and passing through (2, 49).
c
General Equation: f  x  
x
1 a b
Limit to growth = c  c  76
Initial Value = 19  Point (0, 19)
76  19  19a
76
76
f  0 
 19
 19
0
1 a b
1 a
a3
*
Find the logistic function that has an initial value of 19, a
limit to growth of 76, and passing through (2, 49).
Use the point (2, 49) to solve for b:
76
 49
2
1  3b
76  49  147b
3
b
7
76
f  2 
 49
2
1 3 b
Final Answer:
2
f  x 
76
 7
1 3 3
x
*
Determine a formula for the logistic function whose graph
is shown below.
y = 33
(–2, 4)
(0, 6)
Final Answer:
f  x 
33
1  4.5  0.788
x
Use the data below to find an exponential regression for the
population of the U.S., and use this regression to predict the
U.S. population for the year 2000.
Year
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
U.S. Population
Let t = years after 1900
(in millions
Exponential Regression:
76.2
92.2
106.0
123.2
How good is the fit of this model?
132.2
151.3
What about the year 2000?:
179.3
203.3
226.5
(About a 3% overestimate
248.7
of the actual population)
281.4
P t   80.5511.0129
P 100  289.863
t
Use the data below to find logistic regressions for the populations
of FL and PA. Predict the maximum sustainable populations for
these two states. Graph and interpret the regressions.
Populations of Two U.S. States (in millions)
Year
Florida Pennsylvania
1900
0.5
6.3
1910
0.8
7.7
Let t = years after 1800
1920
1.0
8.7
1930
1.5
9.6
1940
1.9
9.9
1950
2.8
10.5
1960
5.0
11.3
1970
6.8
11.8
1980
9.7
11.9
1990
12.9
11.9
2000
16.0
12.3
Use the data below to find logistic regressions for the populations
of FL and PA. Predict the maximum sustainable populations for
these two states. Graph and interpret the regressions.
Population of Florida:
28.021
F t  
0.047015t
1  9018.63e
Population of Pennsylvania:
12.579
P t  
0.034315t
1  29.0003e
Let’s graph them in the window [–10, 300] by [–5, 30]…
The half-life of a certain radioactive substance is 65 days. There
are 3.5 grams present initially. When will there be less than 1 g
remaining?
t 65
1


The Model:
y  3.5  
2


Solve the equation:
where t is time in days
t 65
1
3.5  
2
 1  t  117.478
There will be less than 1 gram
remaining after approximately
117.478 days
The population of deer after t years in Cedar State Park is
modeled by the function
1001
P t  
0.2 t
1  90e
(a) What was the initial population of deer?
P  0  11
(b) When will the number of deer be 600?
1001
Solve graphically: 600 
0.2 t
1  90e
t  24.514yr
(c) What is the maximum number of deer possible in the park?
lim P  t   1001
t 
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Find the logistic function modeling the population that has an
initial population of 25,000, a limit to growth of 500,000, and
a population of 32,000 after 4 years.
c
General Equation: f  x  
x
1 a b
Limit to growth = c  c  500,000
Initial Value = 25,000  Point (0, 25000)
500, 000
500000

25000
1

a



25,
000
0
1 a b
 a  19
*
Find the logistic function modeling the population that has an
initial population of 25,000, a limit to growth of 500,000, and
a population of 32,000 after 4 years.
Plug in (4, 32000):
Final Answer:
f  x 
500, 000
f  4 
 32000
4
1  19 b
500000  32000 1  19b
b  0.937
500,000
1  19  0.937 
x
4

*
Find the logistic function modeling the population that has an
initial population of 8, a limit to growth of 80, and a population
of 60 after 7 years.
c
General Equation: f  x  
x
1 a b
Limit to growth = c  c  80
Initial Value = 8  Point (0, 8)
80
8
0
1 a b
80  81  a 
a9
*
Find the logistic function modeling the population that has an
initial population of 8, a limit to growth of 80, and a population
of 60 after 7 years.
Plug in (7, 60):
80
f 7 
 60
7
1 9 b
80  60 1  9b
Final Answer:
f  x 
b  0.624
80
1  9  0.624
7
x

The 2000 population of Las Vegas, Nevada was 478,000 and
is increasing at the rate of 6.28% each year. At that rate,
when will the population be 1 million?
The Model:
P  t   478000 1.0628 
t
where t is years after 2000
Solve the equation:
478, 000 1.0628   1, 000, 000
 t  12.12
t
In the year 2012, the population
will be 1 million.
Watauga High School has 1200 students. Bob, Carol, Ted, and
Alice start a rumor, which spreads logistically according to the
model below. The model predicts the number of students who
have heard the rumor by the end of t days, where t = 0 is the day
the rumor begins to spread.
1200
S t  
0.9 t
1  39e
1. How many students have heard the rumor by the end of Day 0?
1200
S  0 
 30
1  39
30 students have heard the
rumor on the day the rumor
begins to spread.
Watauga High School has 1200 students. Bob, Carol, Ted, and
Alice start a rumor, which spreads logistically according to the
model below. The model predicts the number of students who
have heard the rumor by the end of t days, where t = 0 is the day
the rumor begins to spread.
1200
S t  
0.9 t
1  39e
2. How long does it take for 1000 students to hear the rumor?
Need to solve the equation:
1200
 1000
0.9 t
1  39e
Solve graphically!!!
 t  5.86
Toward the end of Day 6,
the rumor has reached
the ears of 1000 students