Section 3.5 Exponential and Logarithmic Models

Compound Interest The compound interest formula: A is the amount in the account after t years. P is the principal. r is the annual interest rate. n is the number of pay periods per year.

Example 1 An investment of $5,000 is made into an account that pays 6% annually for 10 years. Find the amount in the account if the interest is compounded: a)annually b)quarterly c)monthly d)daily

a. b. c. d.

Example 2 If interest is compounded semiannually, find how long it will take for $2500 invested at 10% per year to double. P = r = A = n = $ $5000 2

It will take a little more than 7 years.

Compound Continuously After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by A = Pe rt

Example 3 Use the principal, annual interest rate, and years from example 1 but compound continuously. Find the amount in the account.

Example 4 If after 15 years an investment of $100 compounded continuously is worth $300, what is annual interest rate? 300 = 100e 15r 3 = e 15r ln 3 = ln e 15r ln 3 = 15r r ≈ The interest rate is about 7.32%.

Exponential Growth and Decay Formulas 1.Half-Life Formula where N t = the quantity that still remains and has not yet decayed after a time t, N 0 = the initial quantity of the substance that will decay, k = the half-life of the decaying quantity.

2.Exponential Decay Model where b > 0

3.Exponential Growth Model where b > 0

Example 5 Suppose the half-life of a certain radioactive substance is 20 days and there are 5 grams present initially. How much will be left in 70 days? Round to nearest hundredth of a gram. t = N 0 = k = 70 days 5 grams 20 days

Example 6 For a certain strain of bacteria the growth constant (k) = when t is measured in hours. How long will 15 bacteria take to increase to 250 bacteria? N 0 = N t = 250 = 15e 0.658t

It will take more than 4 hours to increase the bacteria to 250 bacteria.

Example 7 According to the U.S. Bureau of the Census in 1990 there were 22.4 million residents of Hispanic origin living in the United States. By 2000, the number had increased to 35.3 million. a.Write the exponential growth equation.

b.Use the model to project the Hispanic population in In 2010 the Hispanic resident population will be about 56.2 million people.

c.In what year will the Hispanic resident population reach 60 million? The Hispanic resident population will reach 60 million people sometime in the year 2011.

Gaussian Model The Gaussian Model is This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve.

For standard normal distributions, the model takes the form The average value for a population can be found from the bell-shaped curve by observing where the maximum y-value of the function occurs. The x-value corresponding to the maximum y-value of the function represents the average value of the independent variable.

Example 8 Last year, the math scores for students in a particular math class roughly followed the normal distribution given by where x is the math score. Sketch the graph of this function and use it to estimate the average math score.

The average score for students in the math class was 74.

Logistic Growth Model Some populations initially have rapid growth, followed by a declining rate of growth. One model for describing this type of growth pattern is the logistic curve given by the function Where y is the population size and x is the time.

An example is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve.

Example 9 On a college campus of 7500 students, one student returns from vacation with a contagious disease and long-lasting virus. The spread of the virus is modeled by where y is the total number of students affected after t days.

The college will cancel classes when 30% or more of the students are infected. a.How many students will be infected after 4 days? Approximately 36 students will be infected.

b.After how many days will the college cancel classes? Classes will be canceled after 9 days.

Logarithmic Model

Example 10 On the Richter scale, the magnitude R of an earthquake of intensity I is given by where I 0 = 1 is the minimum intensity used for comparison. Find the magnitude of R of an earthquake of intensity 68,400,000 to the nearest hundredth.

I = 68,4000