3.5 Exponential & Logarithmic Models

Objective: Use exponential and logarithmic models to solve real life problems.

3.5 Exponential & Logarithmic Models 20141Exponential ModelsExponential Growth Model:

Exponential Decay Model:

Logistic Growth Model:

Logarithmic Models:

2 Exponential Model Exponential Growth/ Decay Model:

3Investment ExampleComplete the table for a savings account in which interest is compounded continuously:

Initial InvestmentAnnual % RateTime to DoubleAmount After 10 Years$20,00010.5%$10,00012 years

4Population Growth ExampleThe population of Higley is given by the exponential equation where P is in thousands and t=0 represents the year 2000. In 1980 the population was 74,000.

Find K, and then use the equation to predict the population in 2020.

5 describes the number of bacteria in a culture after t hours. If n=400 when t=3, find the time required for the bacteria to double.

Bacteria Growth6Modeling Population GrowthA population of fruit flies follows the exponential growth function. After 2 days there are 100 flies and after 4 days there are 300 flies. Find the growth model, and then use it to find out how many flies there will be after 5 days?

7You try:A population of bacteria on your desk follows the exponential growth function (in thousands). After 2 days there are 4 (thousand) bacteria on your desk and after 4 days there are 20 (thousand) bacteria on your desk. Find the growth model, and then use it to find out how much bacteria there will be on your desk in 7 days.

8Radioactive DecayA certain radioactive isotope has a half life of 1200 years . If the initial quantity is 20 grams, use an exponential model to determine how much remains after 1000 years.

9Radioactive DecayThe half-life of radioactive Plutonium is 24,100 years. What percent of a present amount of radioactive strontium will remain after 50,000 years?

10Spread of a VirusOn 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 infected.

How many students are infected after 5 days? After how many days will the college cancel classes?

11SalesThe sales S (in thousands of units) of a cleaning solution after x hundred of dollars is spent on advertising are given by:

When $500 is spent on advertising, 2500 units are sold.Complete the model by solving for k.

Estimate the number of units that will be sold if advertising expenditures are raised to $700.

12Logarithmic ModelOn a Richter scale, the magnitude R of an earthquake of intensity I is given by

where is the minimum intensity used for comparison. Intensity is a measure of the wave energy of an earthquake.

In 2001, the coast of Peru experienced an earthquake that measured 8.4 on the Richter scale. In 2003, Colima, Mexico experienced an earthquake that measured 7.6 on the Richter scale. Find the intensity of each earthquake and compare the two intensities.

13Homework3.5 pg 224 7-13, 19-25, 28, 29, 31 ALL ODDS14