Exponential and Logarithmic Models Section 3.5 Exponential and Logarithmic Models

Objective By following instructions students will be able to: Recognize the five most common types of models involving exponential or logarithmic functions. Use exponential growth and decay functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. Fit exponential and logarithmic models to sets of data.

Exponential Growth Model

Exponential Decay Model

Gaussian Model

Logistic Model

Logarithmic Models

Example 1: Estimates of the world population (in millions) from 1992 through 2000 are shown in the table. The scatter plot of the data is shown in the figure. An exponential growth model that approximates this data is Where P is the population (in millions) and t=2 represents 1992. Compare values given by the model with the estimates given by the U.S. Bureau of the Census. According to this model, what year corresponds to a world population of 6.5 billion? Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 Population 5445 5527 5607 5688 5767 5847 5926 6005 6083

Example 2: In a research experiment, a population of fruit flies is increasing in accordance with the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days?

Age of Dead Organic Material Ratio of the content of radioactive carbon (carbon 14) to the content of nonradioactive carbon (carbon 12) is :

Example 3: The ratio of carbon 14 to carbon 12 in a newly discovered fossil is Estimate the age of the fossil.

Example 4: In 1997, the SAT math scores for college bound seniors roughly followed the normal distribution Where x is the SAT score for mathematics. Use a graphing utility to graph this function. From the graph, estimate the average SAT score.

Example 5: On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by Where y is the total number infected after t days. The college will cancel classes when 40% or more of the students are ill. How many students are infected after 5 days? After how many days will the college cancel classes?

Example 6: On the Richter scale, the magnitude R of an earthquake of intensity I is Where Io=1 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 of energy of the earthquake). Tokyo and Yokohama, Japan in 1924; R=8.3 Izmit, Turkey, in 1999; R=7.4

Example 7: The data in the table gives the yield y (in milligrams) of a chemical reaction after x minutes. Use a graphing utility to fit a logarithmic model to the data. X 1 2 3 4 5 6 7 8 Y 1.5 7.4 10.2 13.4 15.8 16.3 18.2 18.3

Example 8: The total amount A (in billions of dollars) spent on health care in the United States in the years 1970 through 1996 are shown below. Find a model for the data and use the model to estimate the amount spent in 2004. In the list of data points (t,A), t represents the year, with t=0 corresponding to 1970. (Source: U.S. Health Care Financing Administrator) (0,74.3), (182.2), (2,92.3), (3,102.4), (4,115.9), (5,132.6), (6,151.9), (7, 172.6), (8,193.2), (9,218.3), (10, 247.3), (11, 291.4), (12, 328.2), (24, 360.8), (14, 396.0), (15, 428.7), (16, 466.0), (17, 506.2), (18, 562.3), (19, 623.9), (20, 699.5), (21, 766.8), (22, 836.6), (23, 895.1), (24, 945.7), (25, 991.4), (26, 1035.1)

Revisit Objective Did we… Recognize the five most common types of models involving exponential or logarithmic functions? Use exponential growth and decay functions to model and solve real-life problems? Use Gaussian functions to model and solve real-life problems? Use logistic growth functions to model and solve real-life problems? Use logarithmic functions to model and solve real-life problems? Fit exponential and logarithmic models to sets of data?

Homework Pg 266#s 1-41 ODD