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1. Logistic Models Section 3.2b

2. 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!!!

3. Find the logistic function that has an initial value of 5, a limit to growth of 45, and passing through (1, 9). First, recall the general equation: Limit to growth = c Initial Value = 5  Point (0, 5) Finding Logistic Functions

4. 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: Final Answer: Finding Logistic Functions

5. Find the logistic function that has an initial value of 19, a limit to growth of 76, and passing through (2, 49). General Equation: Limit to growth = c Initial Value = 19  Point (0, 19) Finding Logistic Functions

6. 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: Final Answer: Finding Logistic Functions

7. Determine a formula for the logistic function whose graph is shown below. y = 33 (–2, 4) (0, 6) Finding Logistic Functions Final Answer:

8. 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. U.S. Population (in millions Let t = years after 1900 Year Exponential Regression: 1900 76.2 1910 92.2 1920 106.0 1930 123.2 How good is the fit of this model? 1940 132.2 1950 151.3 What about the year 2000?: 1960 179.3 1970 203.3 1980 226.5 (About a 3% overestimate of the actual population) 1990 248.7 2000 281.4

9. 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

10. 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: Population of Pennsylvania: Let’s graph them in the window [–10, 300] by [–5, 30]…

11. 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? The Model: where t is time in days Solve the equation: There will be less than 1 gram remaining after approximately 117.478 days

12. The population of deer after t years in Cedar State Park is modeled by the function (a) What was the initial population of deer? (b) When will the number of deer be 600? Solve graphically: (c) What is the maximum number of deer possible in the park?

13. 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. General Equation: Limit to growth = c Whiteboard Practice  Initial Value = 25,000  Point (0, 25000)

14. 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: Whiteboard practice 

15. 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. General Equation: Limit to growth = c Whiteboard Practice  Initial Value = 8  Point (0, 8)

16. 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): Final Answer: Whiteboard Practice 

17. 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: where t is years after 2000 Solve the equation: In the year 2012, the population will be 1 million.

18. 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. • How many students have heard the rumor by the end of Day 0? 30 students have heard the rumor on the day the rumor begins to spread.

19. 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. 2. How long does it take for 1000 students to hear the rumor? Need to solve the equation: Toward the end of Day 6, the rumor has reached the ears of 1000 students Solve graphically!!!