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**6. 1 Exponential Growth and Decay **

Objectives: Determine the multiplier for exponential growth and decay. Write and evaluate exponential expressions to model growth and decay situations. Standard: A. Analyze a given set of data for the existence of a pattern and represent it algebraically.

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Exponential growth and decay can be used to model a number of real-world situations, such as population growth of bacteria and the elimination of medicine from the bloodstream. You can represent the growth of an initial population of 100 bacteria that doubles every hour by creating a table. The bar chart at the right illustrates how the doubling pattern of growth quickly leads to large numbers. * Time (hr) 1 2 3 4 … n Population 100 200 400 800 1600 100(2)n

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**Find the multiplier for each rate of exponential growth or decay.**

100 • 2n is called an exponential expression because the exponent, n, is a variable and the base, 2, is a fixed number. The base of an exponential expression is commonly referred to as the multiplier. Find the multiplier for each rate of exponential growth or decay. Change percent to correct decimal form. For growth, add the decimal to 1. For decay, subtract the decimal from 1. 7% growth _________ b. 6% decay _________ c. 6.5% growth ______ d. .08% growth_________ e. .05% decay _________ f. 8.2% decay _______

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**II. Modeling Human Population Growth**

Human populations grow much more slowly than bacterial populations. Bacterial populations that double each hour have a growth rate of 100% per hour. The population of the United States in 1990 was growing at a rate of about 8% per decade.

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Ex 2. The population of Brazil was 162,661,000 in 1996 and was projected to grow at a rate of about 7.7% per decade. Predict the population, to the nearest hundred thousand, of Brazil for the years 2016 and 2020. 100% % = 107.7% = 1.077 2016: 2020:

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**III. Modeling Biological Decay**

Caffeine is eliminated from the bloodstream of a child at a rate of about 25% per hour. This exponential decrease in caffeine in a child’s bloodstream is shown in the bar chart. *

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**A rate of decay can be thought of as a negative growth rate.**

To obtain the multiplier for the decrease in caffeine in the bloodstream in a child, subtract the rate of decay from 100%. Thus the multiplier is 0.75.

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**100% - 15% = 85% = .85 30(.85)x 1 Hour: 30(.85)1 = 25.5 mg 4 Hours:**

Ex 1. The rate at which caffeine is eliminated from the bloodstream of an adult is about 15% per hour. An adult drinks a caffeinated soda, and the caffeine in his or her bloodstream reaches a peak level of 30 milligrams. Predict the amount, to the nearest tenth of a milligram, of caffeine remaining 1 hour after the peak level and 4 hours after the peak level. 100% - 15% = 85% = .85 30(.85)x 1 Hour: 30(.85)1 = 25.5 mg 4 Hours: 30(.85)4 = 15.7 mg

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**100% - 20% = 80% = .80 300(.8)x 2 Hours: 7 hours:**

Ex 2. A vitamin is eliminated from the bloodstream at a rate of about 20% per hour. The vitamin reaches a peak level in the bloodstream of 300 milligrams. Predict the amount, to the nearest tenth of a milligram, of vitamin remaining 2 hour after the peak level and 7 hours after the peak level. 100% - 20% = 80% = .80 300(.8)x 2 Hours: 7 hours:

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**evaluate each expression. 1). 2x 2). 3y**

** Given x = 5, y =3/5, z = 3.3, evaluate each expression. 1). 2x ). 3y 3). 50 (2)3x ). 10(2) z + 2

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Writing Journal:

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Lesson Quiz

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Logarithms and Exponential Models Lesson 4.2. Using Logarithms Recall our lack of ability to solve exponential equations algebraically We cannot manipulate.

Logarithms and Exponential Models Lesson 4.2. Using Logarithms Recall our lack of ability to solve exponential equations algebraically We cannot manipulate.

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