Exponential Exponential ModelingModeling

Section 3.2aSection 3.2a

Let’s start with a whiteboard problem today…

Determine a formula for the exponential function whose graph isshown below.

(0,3) (4, 1.49)

0xf x f b

000 3f f b 0 3f

44 3 1.49f b

41.49

3b 0.839

3 0.839xf x

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

Time in years Population

0

1

2

3

t

00P P Initial population

0 01P P P r 0 1P r 2 1 1P P r 20 1P r 3 2 1P P r 30 1P r

0 1t

P t P r

Exponential Population ModelIf a population P is changing at a constant percentagerate r each year, then

0 1t

P t P r where P is the initial population, r is expressed as adecimal, and t is time in years.

0

Exponential Population Model

If r > 0, then P( t ) is an exponential growth function, and itsgrowth factor is the base: (1 + r).

0 1t

P t P r

Growth Factor = 1 + Percentage RateGrowth Factor = 1 + Percentage Rate

If r < 0, then P( t ) is an exponential decay function, and itsdecay factor is the base: (1 + r).

Decay Factor = 1 + Percentage RateDecay 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.

782,248 1.0135tP t 1.

1 + r = 1.0135 r = 0.0135 > 0

P is an exp. growth func. with a growth rate of 1.35%

1,203,368 0.9858tP t 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.

0 12P 8% 0.08r

12 1 0.08t

P t 12 1.08t

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.

200 100 2 First, create the model:

Total bacteria after 1 hour:2400 100 2 Total bacteria after 2 hours:

3800 100 2 Total bacteria after 3 hours:

100 2tP t 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:

11.77t Interpret:

The population of the bacteria will beThe population of the bacteria will be350,000 in about 11 hours and 46 minutes350,000 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 decayradioactive decay.

The half-lifehalf-life of 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.

20 2012.5 5 2First, create the model:

Grams after 20 days:

2015 2

t

f t Grams after t days:

40 2011.25 5 2Grams 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:

46.44t Interpret:

There will be 1 gram of the radioactive substanceThere will be 1 gram of the radioactive substanceleft after approximately 46.44 days (46 days, 11 hrs)left after approximately 46.44 days (46 days, 11 hrs)