Chapter 6

Exponential and Logarithmic Functions

EXPONENTIAL GROWTH AND DECAY

Section 6.1

Modeling Bacterial GrowthYou can use a calculator to model the growth of 25 bacteria, assuming that the entire population doubles every hour. Copy and complete the table below:

Time (hr) 0 1 2 3 4 5 6

Population 25 50

Modeling Bacterial GrowthYou can represent the growth of an initial population of 100 bacteria that doubles every hour by completing the table below:

The population after n hours can be represented by the following exponential expression:

n times

Time (hr) 0 1 2 3 4 5 6

Population 100

Vocabulary

• The expression, 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.

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 U.S. in 1990 was growing at a rate of about 8% per decade.

Modeling Human Population Growth

The population of the U.S. was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

1. To obtain the multiplier for exponential growth, add the growth rate to 100%.

2. Write the expression for the population n decades after 1990.

3. How many years past 1990 is 2010? How many decades is this?

4. How many years past 1990 is 2010? How many decades is this?

5.

Deeper Thinking…• Why would we ADD to 100% for an exponential

growth?• What does growth mean?• Don’t forget to always change your percentages to

decimals and then use this as the multiplier.• What would it mean for a population to have a

growth rate of 100%?– Ex: Population = 300 100% growth = ?– What would a 200% growth rate look like for this

population?– Would 100% change the population?

• Why do we ADD the growth rate to 100%?

Modeling Biological Decay

• Caffeine is eliminated from the bloodstream of a child at a rate of about 25% per hour.

• A rate of decay can be though of as a negative growth rate.

Modeling Biological DecayThe rate at which caffeine is eliminated from the bloodstream of an adult is about 15% per hour. An adult drinks 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.

1. To obtain the multiplier for exponential decay, subtract the growth rate to 100%.

2. Write the expression for the caffeine level x hours after the peak level.

3. Find the remaining caffeine amount after 1 hour.

4. Find the remaining caffeine amount after 4 hours.

5.

Deeper Thinking…• Why would we SUBTRACT from 100% for an

exponential decay?• What does the word decay mean?• Don’t forget to always change your percentages

to decimals and then use this as the multiplier.• What would it mean for a population to have a

decay rate of 100%?– Ex: Population = 300 100% decay = ?– What would a 25% decay rate look like for this

population?• Why do we SUBTRACT the decay rate from 100%?

PracticeFind the multiplier ofr each rate of exponential growth or decay.

1. 7% growth

2. 2% decay

3. 0.05% decay

4. 9% growth

5. 8.2% decay

6. 0.075% growth

Given x = 5, y = 3/5, and z = 3.3, evaluate each expression.

7. 2x

8. 50(2)3x

9. 3y

10. 25(2)z

11. 10(2)z+2

PracticePredict the population of bacteria for each situation and time period (be sure to use the formula and show all work!!).

1. 55 bacteria that double every houra. after 3 hours

b. after 5 hours

2. 75 E.coli bacteria that double every 30 minutesa. after 2 hours

b. after 3 hours

c.

Homework

Finish worksheet 6.1