3 2 Exponential and Logistic Modeling Exponential Population

3. 2 Exponential and Logistic Modeling

Exponential Population Model If a populations P is changing at a constant percentage rate r each year, then P(t) = P 0 (1 + r)t P 0 = Initial Population r = rate expressed as a decimal t = time in years

Growth and Decay Factors P(t) = P 0 (1 + r)t If r > 0, then P(t) is an exponential growth function Growth Factor = 1 + r If r < 0, then P(t) is an exponential decay function Decay Factor = 1 + r

Finding growth and decay rates: Exponential growth or decay function? Find constant percentage rate of growth/decay. f(x) = 78, 963 0. 968 x g(t) = 247 2 t You Try! P(t) = 3. 5 1. 09 t

Finding an Exponential Function: Initial population = 28, 900, decreasing at a rate of 2. 6% per year You Try! Initial Height = 18 cm, growing at a rate of 5. 2% per week

Find the logistic function that satisfies the given conditions: Initial Value = 10; Limit to Growth = 40; Passing through (1, 20)

You Try! Find the logistic function that satisfies the given conditions: Initial Population = 16; Maximum Sustainable Population = 128; Passing through (5, 32)

You Try! Exponential Growth: The population of Smallville in the year 1890 was 6250. Assume the population increased at a rate of 2. 75% per year. a) Estimate the population in 1915 and 1940 a) Predict when the population reached 50, 000

Radioactive Decay: The half-life of a certain radioactive substance is 14 days. There are 6. 6 grams initially present. Express that amount of substance remaining as a function of time t When will there be less than 1 gram remaining?

Homework Pg. 296 -299 (4, 8, 16, 24, 32, 42) Chapter 3 Vocabulary Sheet Due Tomorrow