Exponential ModelingSection 3.2a

Lets start with a whiteboard problem todayDetermine a formula for the exponential function whose graph isshown below.(0,3)(4, 1.49)

Constant Percentage RatesIf r is the constant percentage rate of change of a population,then the population follows this pattern:Time in yearsPopulation0123t Initial population

Exponential Population ModelIf a population P is changing at a constant percentagerate r each year, thenwhere P is the initial population, r is expressed as adecimal, and t is time in years.0

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

Finding Growth and Decay RatesTell whether the population model is an exponential growthfunction or exponential decay function, and find the constantpercentage 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 FunctionDetermine the exponential function with initial value = 12,increasing at a rate of 8% per year.

Modeling: Bacteria GrowthSuppose a culture of 100 bacteria is put into a petri dish andthe culture doubles every hour. Predict when the number ofbacteria 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 GrowthSuppose a culture of 100 bacteria is put into a petri dish andthe culture doubles every hour. Predict when the number ofbacteria will be 350,000.Now, solve graphically to find where the population functionintersects y = 350,000:Interpret:The population of the bacteria will be350,000 in about 11 hours and 46 minutes

Modeling: Radioactive DecayWhen an element changes from a radioactive state to anon-radioactive state, it loses atoms as a fixed fraction perunit time Exponential Decay!!!This process is called radioactive decay.The half-life of a substance is the time ittakes for half of a sample of the substanceto change state.

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

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

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