Exponential Modeling Section 3.2a

Let’s start with a whiteboard problem today… Determine a formula for the exponential function whose graph is shown below. (4, 1.49) (0,3)

Constant Percentage Rates If r is the constant percentage rate of change of a population, then the population follows this pattern: Time in years Population Initial population 1 2 3 t

Exponential Population Model If a population P is changing at a constant percentage rate r each year, then where P is the initial population, r is expressed as a decimal, and t is time in years.

Exponential Population Model If r > 0, then P( t ) is an exponential growth function, and its growth factor is the base: (1 + r). Growth Factor = 1 + Percentage Rate If r < 0, then P( t ) is an exponential decay function, and its decay factor is the base: (1 + r). Decay Factor = 1 + Percentage Rate

Finding Growth and Decay Rates Tell whether the population model is an exponential growth function or exponential decay function, and find the constant percentage rate of growth or decay. 1. 1 + r = 1.0135  r = 0.0135 > 0  P is an exp. growth func. with a growth rate of 1.35% 2. 1 + r = 0.9858  r = –0.0142 < 0  P is an exp. decay func. with a decay rate of 1.42%

Finding an Exponential Function Determine the exponential function with initial value = 12, increasing at a rate of 8% per year.

Modeling: Bacteria Growth Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. First, create the model: Total bacteria after 1 hour: Total bacteria after 2 hours: Total bacteria after 3 hours: Total bacteria after t hours:

Modeling: Bacteria Growth Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. Now, solve graphically to find where the population function intersects y = 350,000: Interpret: The population of the bacteria will be 350,000 in about 11 hours and 46 minutes

Modeling: Radioactive Decay When an element changes from a radioactive state to a non-radioactive state, it loses atoms as a fixed fraction per unit time  Exponential Decay!!! This process is called radioactive decay. The half-life of a substance is the time it takes for half of a sample of the substance to change state.

Modeling: Radioactive Decay Suppose the half-life of a certain radioactive substance is 20 days and there are 5 grams present initially. Find the time when there will be 1 gram of the substance remaining. First, create the model: Grams after 20 days: Grams after 40 days: Grams after t days:

Modeling: Radioactive Decay Suppose the half-life of a certain radioactive substance is 20 days and there are 5 grams present initially. Find the time when there will be 1 gram of the substance remaining. Now, solve graphically to find where the function intersects the line y = 1: Interpret: There will be 1 gram of the radioactive substance left after approximately 46.44 days (46 days, 11 hrs)