Exponential Growth & DecayExponential Growth & DecaySection 4.5Section 4.5

JMerrill, 2005JMerrill, 2005

Revised 2008Revised 2008

ReviewReview

3 3log log 6 3x x

3 3

3

log log 6 3

log 6 3

x x

x x

3 3

3

3

log log 6 3

log 6 3

3 6

x x

x x

x x

3 3

3

3

2

log log 6 3

log 6 3

3 6

27 6

x x

x x

x x

x x

Is this okay?

Arguments must be positive

3 3

3

3

2

2

log log 6 3

log 6 3

3 6

27 6

6 27 0

x x

x x

x x

x x

x x

3 3

3

3

2

2

log log 6 3

log 6 3

3 6

27 6

6 27 0

9 3 0

x x

x x

x x

x x

x x

x x

3 3

3

3

2

2

log log 6 3

log 6 3

3 6

27 6

6 27 0

9 3 0

9, 3

x x

x x

x x

x x

x x

x x

x

9, 3x 9x

ReviewReview

500e0.3x = 600

e0.3x = 1.2

ln 1.2 = 0.3x

ln1.2

0.3x

x = 0.608

Exponents and LogarithmsExponents and Logarithms How are exponents and logarithms related? How are exponents and logarithms related?

They are inverses of each otherThey are inverses of each other Why is this important?Why is this important? Using inverses allow us to solve problems Using inverses allow us to solve problems

(we use subtraction to solve addition (we use subtraction to solve addition problems & division to solve multiplication)problems & division to solve multiplication)

Many real-life scenarios are exponential in Many real-life scenarios are exponential in nature and logarithms allow us to solve for nature and logarithms allow us to solve for the unknown.the unknown.

Examples Using Logarithmic ScalesExamples Using Logarithmic Scales

The Richter scale is used to determine the The Richter scale is used to determine the intensity of an earthquake. intensity of an earthquake.

Measuring acidity using the pH scale, or Measuring acidity using the pH scale, or concentration of ions.concentration of ions.

Carbon dating. Carbon dating. Modeling population growth/decay--just to Modeling population growth/decay--just to

name a few…name a few…

Exponential Decay ModelExponential Decay Model

A(t) = AA(t) = A00eektkt

AA0 0 is the initial amountis the initial amount K is the growing/decay entity. If k>0, the K is the growing/decay entity. If k>0, the

entity is growing (an increasing function). If entity is growing (an increasing function). If k<0, the entity is decaying (a decreasing k<0, the entity is decaying (a decreasing function).function).

Looks like A(t) = PeLooks like A(t) = Pertrt? It works the same ? It works the same way. way.

Population ModelPopulation Model

In 1970, the US population was 203.3 In 1970, the US population was 203.3 million. In 2003, the population was 294 million. In 2003, the population was 294 million. million.

1.1. Find the exponential growth modelFind the exponential growth model

2.2. By which year will the US population reach By which year will the US population reach 315 million?315 million?

Population ModelPopulation Model

t is the number of years after 1970. t is the number of years after 1970. t=0 represents 1970. t = 33 represents 2003t=0 represents 1970. t = 33 represents 2003 When t = 33, A = 294 When t = 33, A = 294

A(t) = AA(t) = A00eektkt

294 = 203.3e294 = 203.3ek(33)k(33)

Population Con’tPopulation Con’t

294 = 203.3e294 = 203.3ek(33)k(33)

33k294ln lne

203.3

33k294e

203.3

294ln 33klne

203.3

What do you do when the

exponent is a variable?

What does lne = ?

294203.3 0.011ln k

33

So, k ≈ 0.011, which is exponential growth

The growth model is

A(t) = 203.3eA(t) = 203.3e0.011t0.011t

Population Con’tPopulation Con’t

When will the population reach 315 million?When will the population reach 315 million? A(t) = 203.3eA(t) = 203.3e0.011t0.011t

315 = 203.3e315 = 203.3e0.011t0.011t

You finish…You finish…

Did you get approximately 40? Did you get approximately 40? That means that in the year 2010 the That means that in the year 2010 the

population will be approx. 315 million!population will be approx. 315 million!

Carbon DatingCarbon Dating

The natural base, e, is The natural base, e, is used to estimate the ages used to estimate the ages of artifacts. Plants and of artifacts. Plants and animals absorb Carbon-14 animals absorb Carbon-14 from the atmosphere. from the atmosphere. When a plant or animal When a plant or animal dies, the amount of dies, the amount of carbon-14 it contains carbon-14 it contains decays in such a way that decays in such a way that exactly half of the initial exactly half of the initial amount is present after amount is present after 5,715 years. 5,715 years.

Carbon DatingCarbon Dating

The function that The function that models the decay of models the decay of carbon-14, where Acarbon-14, where A00 is is

the initial amount of the initial amount of carbon-14, and A(t) is carbon-14, and A(t) is the amount present the amount present tt years after the plant or years after the plant or animal dies, isanimal dies, is

0( ) ktA t A e

Carbon Dating ExampleCarbon Dating Example

Archaeologists find scrolls and claim that Archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original the scrolls contain 78% of their original carbon-14. Could the scrolls be 2000 years carbon-14. Could the scrolls be 2000 years old?old?

Using the same process as the last Using the same process as the last example, we find k to be -0.00012. example, we find k to be -0.00012.

Finding k is written out in the book on P449. Finding k is written out in the book on P449.

Carbon Dating ExampleCarbon Dating Example

0.000120

0.000120 0

0.00012

( )

.78

.78

ln.78 .00012 ln

2070.5

t

t

t

A t A e

A A e

e

t e

t

78% of the original amount

You DoYou Do

A wooden chest is found and said to be from A wooden chest is found and said to be from the 2the 2ndnd century BCE. Tests on a sample of century BCE. Tests on a sample of wood from the chest reveal that it contains wood from the chest reveal that it contains 92% of its original carbon-14. Could the 92% of its original carbon-14. Could the chest be from the 2chest be from the 2ndnd century BCE? century BCE?

Use the same k as the last example.Use the same k as the last example.

You doYou do0.00012

0

0.000120 0

0.00012

( )

.92

.92

ln.92 .00012 ln

694.85

t

t

t

A t A e

A A e

e

t e

t

Logistic Growth ModelLogistic Growth Model

The spread of disease is exponential in The spread of disease is exponential in nature. However, there aren’t an infinite nature. However, there aren’t an infinite number of people. Eventually, the disease number of people. Eventually, the disease has to level off. Growth is always limited. A has to level off. Growth is always limited. A logistic growth model is used in this type of logistic growth model is used in this type of situation:situation:

Y = c is the horizontal asymptote. Thus c is Y = c is the horizontal asymptote. Thus c is the limiting value of the function.the limiting value of the function.

bt

cf(t)

1 ae

Modeling the Spread of the FluModeling the Spread of the Flu

The function below describes the number of The function below describes the number of people, f(people, f(tt), who have become ill with ), who have become ill with influenza influenza tt weeks after its initial outbreak in weeks after its initial outbreak in a town with a population of 30,000 people.a town with a population of 30,000 people.

1.5t

30,000f(t)

1 20e

Modeling the Spread of the FluModeling the Spread of the Flu

1.1. How many people became ill with the flu when How many people became ill with the flu when the epidemic began?the epidemic began?

2.2. How many people were ill by the end of the fourth How many people were ill by the end of the fourth week?week?

3.3. What is the limiting size of f(t), the population that What is the limiting size of f(t), the population that become ill?become ill?

1.5t

30,000f(t)

1 20e

Modeling the Spread of the FluModeling the Spread of the Flu

1.1. How many people became ill with the flu How many people became ill with the flu when the epidemic began?when the epidemic began?

In the beginning, t = 0:In the beginning, t = 0:

1.5(0)

30,000f(t)

1 20e

30,000f(t) 1429

1 20

Modeling the Spread of the FluModeling the Spread of the Flu

2. How many people were ill by the end of the 2. How many people were ill by the end of the fourth week? fourth week?

1.5(4)

30,000f(t)

1 20e

f (t) 28,583

Modeling the Spread of the FluModeling the Spread of the Flu

3. What is the limiting size of f(t), the 3. What is the limiting size of f(t), the population that become ill?population that become ill?

C represents the limiting size that f(t) can obtain. There are only 30,000 people in the town, therefore, the limiting size must be 30,000!