Lesson 5.5: Exponential and Logarithmic Models Five Most Common - - PowerPoint PPT Presentation

y ae b

bx

  ,

Lesson 5.5: Exponential and Logarithmic Models

Five Most Common Models

• 1. Exponential Growth model:
• 2. Exponential Decay model:
• 3. Gausian model:
• 4. Logistic Growth model:
• 5. Logarithmic model:

y ae b

bx

 

 ,

y ae

x b c



 

b g

2 /

y a be rx  



1 y a b x y a b x     ln , log10

Ex 1: Exponential Growth

In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days, there are 300 flies. How many flies are there after 5 days?

y aebx 

x = time, y = # of flies

y aebx 

x = time, y = # of flies

100

2

 ae b

a e b  100

2

300

4

 ae b

300 100

2 4

F

H I K

e e

b b

300 100 2  e b e b

2

3  ln ln e b

2

3  2 3 b  ln

b  ln3 2

a  100 3  33

y e

x

 33 0 549

.

y e  33

0 549 5 .

b g

y flies  514

 0549 .

Ex 2: Exponential Decay

In living organic material, the ratio of the number of radioactive carbon isotopes (carbon-14) to the non- radioactive carbon isotopes (carbon-12) is about 1 to

• 1012. When organic material dies, its carbon-12 content

remains fixed while the carbon-14 begins to decay with a half-life of 5700 years. To estimate the age of dead

• rganic material, scientists us the following formula, which

denotes the ratio of carbon-14 to carbon-12 present at any time t (in years)

R e t 



1 1012

8223 /

Find the age of a newly discovered fossil if the ratio of carbon-14 to carbon-12 is

1 1013

Half-life of Carbon-14 = 5715 years. How long until only 25% of fossil is remaining?

Ex 2: Continued.

y aekx  .5

5715

C Ce

k

 .5

5715

 e

k

ln . ln 5

5715

b g c h

 e

k

5715 5 k  ln .

b g

k  ln .5 5715

b g

 000012 .

.

.

25

0 00012

C Ce

x





.

.

25

0 00012





e

x

ln . ln

.

25

0 00012

b g c h





e

x

  000012 25 . ln . x

b g

x   ln . . 25 000012

b g

x years 11552 ,

Ex 3: Logarithmic Model

On the Richter Scale, the magnitude, R, of an earthquake intensity, I, is

R I I  log10

where I0 is the minimum intensity used for comparison. Find the intensities per unit of area for the following

• earthquakes. (Intensity is a measure of the wave

energy of an earthquake.)

• A. Tokyo and Yokohama, Japan 1923: R = 8.3
• B. El Salvador, 2001: R = 7.7

Ex 3: Continued.

Homework: p.416-417 #37, 39, 42, 51, 52

8 3 1

10

. log  I 83

10

. log  I 10 10

8 3

10

. log



I

I 108 3

.

7 7 1

10

. log  I 7 7

10

. log  I 10 10

7 7

10

. log



I

I 107 7

.

 3981 .

Conclusion: The 1923 earthquake had an intensity approximately four times greater than the 2001 earthquake

I 199 526 2315 , , . I  50118 7234 , , .

199 526 2315 50118 7234 , , . , , .