1. Constant Rate Exponential Population Model Date: 3.2 Exponential and Logistic Modeling (3.2) Find the growth or decay rates: r = (1 + r) 1.35% growth If a population P is changing at a constant percentage rate r each year, then P(t) = P 0 (1 + r) t Where P 0 is the initial population, r is expressed as a decimal, and t is time in years. If a population P is changing at a constant percentage rate r each year, then P(t) = P 0 (1 + r) t Where P 0 is the initial population, r is expressed as a decimal, and t is time in years. (1 + r) 1.42% decay Determine the exponential function with initial value = 12, increasing at a rate of 8 % per year?

2. Modeling bacteria growth Suppose a culture of 100 bacteria is put in a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. doubles means what %? what is the initial population? represents the bacteria population t hours after placed in the petri dish. Use the graphing calculator to calculate the intersection when y = 350,000 Intersection: (11.773139, 350000) 11 hr + (.773139)60 min The number of bacteria will be 350,000 in 11 hours and 46 minutes.

3. Modeling radioactive decay Suppose the half-life of a certain radioactive substance is 20 days and there are 5 g present initially. Find the time when there will be 1 g of the substance remaining. half-life means what %? not daily, every 20 days means? Use the graphing calculator to calculate the intersection when y = 1 Intersection: (46.438562, 1) 46 days + (.438562)24 hrs There will be 1 g of the radioactive substance remaining after approximately 46 days and 11 hours. t per 20 days Initial population? Models the mass in grams of the radioactive substance at time t days.

4. DAY 2 Modeling using exponential regression Use the 1900-1990 data table and exponential regression to predict the U.S. population for 2000. Compare the result with the listed value for 2000. Use STAT to enter data in L 1 and L 2 Do not enter data for year 2000 Graph using STAT PLOT.

5. Modeling using exponential regression Use the 1900-1990 data table and exponential regression to predict the U.S. population for 2000. Compare the result with the listed value for 2000. Use STAT then CALC to write an exponential regression model for the data The exponential regression predicts the population will be 294.3 million. Use the TRACE to calculate the population for the year 2000. Compare the result with the listed value for 2000. 294.3 – 281.4 = 12.9 It is overestimated by 12.9 million, less than a 5 % error.

6. Modeling using logistic regression Use the data table and logistic regression to predict the maximum sustainable populations for Florida and Pennsylvania. Use STAT to enter data in L 1, L 2 and L 3 Graph using STAT PLOT. Use different symbols for each graph. [0,210] by [-5,20]

7. Modeling using logistic regression Use the data table and logistic regression to predict the maximum sustainable populations for Florida and Pennsylvania. Use STAT then CALC to write a logistic model for the data Predict the maximum sustainable populations: The maximum sustainable population for Florida is about 28.0 million and for Pennsylvania is about 12.6 million.

8. Exponential and Logistic Modeling