Section 3.2bSection 3.2b

In the last section, we did plenty ofIn the last section, we did plenty ofanalysisanalysis of logistic functions that of logistic functions that

were were givengiven to us… to us…

Now, we begin work onNow, we begin work onfinding our very ownfinding our very ownlogistic functions!!!logistic functions!!!

1 x

cf x

a b

Find the logistic function that has an initial value of 5, alimit to growth of 45, and passing through (1, 9).

First, recall the general equation:

Limit to growth = c 45c Initial Value = 5 Point (0, 5)

0

450 5

1f

a b

45

51 a

45 5 5a

8a

1

451 9

1 8f

b

Find the logistic function that has an initial value of 5, alimit to growth of 45, and passing through (1, 9).

Use the point (1, 9) to solve for b:

459

1 8b

45 9 72b 1

2b

Final Answer:Final Answer:

45

11 8 2xf x

1 x

cf x

a b

Find the logistic function that has an initial value of 19, alimit to growth of 76, and passing through (2, 49).

General Equation:

Limit to growth = c 76c Initial Value = 19 Point (0, 19)

0

760 19

1f

a b

76

191 a

76 19 19a

3a

2

762 49

1 3f

b

Find the logistic function that has an initial value of 19, alimit to growth of 76, and passing through (2, 49).

Use the point (2, 49) to solve for b:

2

7649

1 3b

276 49 147b

3

7b

Final Answer:Final Answer:

76

31 3 7

xf x

Determine a formula for the logistic function whose graphis shown below.

(0, 6)(–2, 4)

y = 33

Final Answer:Final Answer: 33

1 4.5 0.788xf x

Use the data below to find an exponential regression for thepopulation of the U.S., and use this regression to predict theU.S. population for the year 2000.

YearU.S. Population

(in millions1900 76.21910 92.21920 106.01930 123.21940 132.21950 151.31960 179.31970 203.31980 226.51990 248.72000 281.4

Exponential Regression:

80.551 1.0129tP t

Let t = years after 1900

How good is the fit of this model?

What about the year 2000?:

100 289.863P (About a 3% overestimateof the actual population)

Use the data below to find logistic regressions for the populationsof FL and PA. Predict the maximum sustainable populationsmaximum sustainable populations forthese two states. Graph and interpret the regressions.

YearPopulations of Two U.S. States (in millions)

1900 0.51910 0.81920 1.01930 1.51940 1.91950 2.81960 5.01970 6.81980 9.71990 12.92000 16.0

Florida Pennsylvania6.37.78.79.69.910.511.311.811.911.912.3

Let t = years after 1800

Use the data below to find logistic regressions for the populationsof FL and PA. Predict the maximum sustainable populationsmaximum sustainable populations forthese two states. Graph and interpret the regressions.

Population of Florida:

0.047015

28.021

1 9018.63 tF t

e

Population of Pennsylvania:

0.034315

12.579

1 29.0003 tP t

e

Let’s graph them in the window [–10, 300] by [–5, 30]…Let’s graph them in the window [–10, 300] by [–5, 30]…

The half-life of a certain radioactive substance is 65 days. Thereare 3.5 grams present initially. When will there be less than 1 gremaining? 65

13.5

2

t

y

The Model:

There will be less than 1 gramThere will be less than 1 gramremaining after approximatelyremaining after approximately

117.478 days117.478 days

where t is time in days

Solve the equation:

117.478t 65

13.5 1

2

t

The population of deer after t years in Cedar State Park ismodeled by the function

0.2

1001

1 90 tP t

e

(a) What was the initial population of deer? 0 11P (b) When will the number of deer be 600?

Solve graphically:0.2

1001600

1 90 te

24.514yrt

(c) What is the maximum number of deer possible in the park?

lim 1001tP t

1 x

cf x

a b

Find the logistic function modeling the population that has aninitial population of 25,000, a limit to growth of 500,000, anda population of 32,000 after 4 years.

General Equation:

Limit to growth = c 500,000c Initial Value = 25,000 Point (0, 25000)

0

500,00025,000

1 a b

500000 25000 1 a

19a

4

500,0004 32000

1 19f

b

Find the logistic function modeling the population that has aninitial population of 25,000, a limit to growth of 500,000, anda population of 32,000 after 4 years.

Plug in (4, 32000):

4500000 32000 1 19b

0.937bFinal Answer:Final Answer:

500,000

1 19 0.937xf x

1 x

cf x

a b

Find the logistic function modeling the population that has aninitial population of 8, a limit to growth of 80, and a populationof 60 after 7 years.

General Equation:

Limit to growth = c 80c Initial Value = 8 Point (0, 8)

0

808

1 a b

80 8 1 a 9a

7

807 60

1 9f

b

Find the logistic function modeling the population that has aninitial population of 8, a limit to growth of 80, and a populationof 60 after 7 years.

Plug in (7, 60):

780 60 1 9b

0.624bFinal Answer:Final Answer:

80

1 9 0.624xf x

The 2000 population of Las Vegas, Nevada was 478,000 andis increasing at the rate of 6.28% each year. At that rate,when will the population be 1 million?

478000 1.0628t

P t The Model:

In the year 2012, the populationIn the year 2012, the populationwill be 1 million.will be 1 million.

Solve the equation:

12.12t 478,000 1.0628 1,000,000

t

where t is years after 2000

Watauga High School has 1200 students. Bob, Carol, Ted, andAlice start a rumor, which spreads logistically according to themodel below. The model predicts the number of students whohave heard the rumor by the end of t days, where t = 0 is the daythe rumor begins to spread.

0.9

1200

1 39 tS t

e

1. How many students have heard the rumor by the end of Day 0?

12000 30

1 39S

30 students have heard the30 students have heard therumor on the day the rumorrumor on the day the rumor

begins to spread.begins to spread.

Watauga High School has 1200 students. Bob, Carol, Ted, andAlice start a rumor, which spreads logistically according to themodel below. The model predicts the number of students whohave heard the rumor by the end of t days, where t = 0 is the daythe rumor begins to spread.

0.9

1200

1 39 tS t

e

2. How long does it take for 1000 students to hear the rumor?

0.9

12001000

1 39 te

Toward the end of Day 6,Toward the end of Day 6,the rumor has reachedthe rumor has reached

the ears of 1000 studentsthe ears of 1000 students

Need to solve the equation:

Solve graphically!!!

5.86t