1. Chapter 10.6 Exponentials Growth and Decay Standard & Honors
Algebra II Mr. Gilbert
Chapter 10.6
Exponentials Growth and Decay
Standard & Honors
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2. Agenda Warm up Work book (participation grade) Lesson Homework
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3. Click the mouse button or press the Space Bar to display the answers.
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Click the mouse button or press the Space Bar to display the answers.
Transparency 6

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Transparency 6a

5. Homework Review
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6. Communicate Effectively
Decay Equation
Growth Equation
Natural base (e)
Common base (10)
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7. (4) Example 1 Exponential Decay of the Form y = a(1 – r)t
Example 2 Exponential Decay of the Form y = ae–kt(5)
Example 3 Exponential Growth of the Form y = a(1 + r )t (4)
Example 4 Exponential Growth of the Form y = aekt (3)
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Lesson 6 Contents

8. Caffeine A cup of coffee contains 130 milligrams of caffeine
Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 90% of this caffeine to be eliminated from a person’s body?
Explore
The problem gives the amount of caffeine consumed and the rate at which the caffeine is eliminated. It asks you to find the time it will take for 90% of the caffeine to be eliminated from a person’s body.
Use the formula Let t be the number of hours since drinking the coffee. The amount remaining y is 10% of 130 or 13.
Plan
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Example 6-1a

9. Exponential decay formula
Solve
Exponential decay formula
Replace y with 13, a with 130, and r with 0.11.
Divide each side by 130.
Property of Equality for Logarithms
Power Property for Logarithms
Divide each side by log 0.89.
Use a calculator.
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Example 6-1b

10. Exponential decay formula
Answer: It will take approximately 20 hours for 90% of the caffeine to be eliminated from a person’s body.
Examine
Use the formula to find how much of the original 130 milligrams of caffeine would remain after 20 hours.
Exponential decay formula
Replace a with 130, r with and t with 20.
Ten percent of 130 is 13, so the answer seems reasonable.
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Example 6-1c

11. Caffeine A cup of coffee contains 130 milligrams of caffeine
Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 80% of this caffeine to be eliminated from a person’s body?
Answer: hours
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Example 6-1d

12. Exponential decay formula
Geology The half-life of Sodium-22 is 2.6 years. What is the value of k for Sodium-22?
Exponential decay formula
Replace y with 0.5a and t with 2.6.
Divide each side by a.
Property of Equality for Logarithmic Functions
Inverse Property of Exponents and Logarithms
Divide each side by –2.6.
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Example 6-2a

13. Use a calculator.
Answer: The constant k for Sodium-22 is Thus, the equation for the decay of Sodium-22 is where t is given in years.
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Example 6-2b

14. Formula for the decay of Sodium-22
A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth?
Formula for the decay of Sodium-22
Replace y with 0.1a.
Divide each side by a.
Property of Equality for Logarithms
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Example 6-2c

15. Inverse Property for Exponents and Logarithms
Divide each side by –
Use a calculator.
Answer: It was formed about 9 years ago.
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Example 6-2d

16. a. What is the value of k for radioactive iodine?
Health The half-life of radioactive iodine used in medical studies is 8 hours.
a. What is the value of k for radioactive iodine?
b. A doctor wants to know when the amount of radioactive iodine in a patient’s body is 20% of the original amount. When will this occur?
Answer:
Answer: about 19 hours later
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Example 6-2e

17. Multiple-Choice Test Item The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?
A 4 years B 5 years
C 20 years D 23 years
Read the Test Item
You want to know when the population has doubled or is 2 million. Use the formula
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Example 6-3a

18. Exponential growth formula
Solve the Test Item
Exponential growth formula
Replace y with 2,000,000, a with 1,000,000, and r with 0.03.
Divide each side by 1,000,000.
Property of Equality for Logarithms
Power Property of Logarithms
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Example 6-3b

19. Divide each side by ln 1.03. Use a calculator. Answer: D 11/24/2018
Example 6-3c

20. Multiple-Choice Test Item The population of a city of 10,000 is increasing at a rate of 5% per year. If the population continues to grow at this rate, in how many years will the population have doubled?
A 10 years B 12 years
C 14 years D 18 years
Answer: C
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Example 6-3d

21. You want to find t such that
Population As of 2000, Nigeria had an estimated population of 127 million people and the United States had an estimated population of 278 million people. The growth of the populations of Nigeria and the United States can be modeled by and , respectively. According to these models, when will Nigeria’s population be more than the population of the United States?
You want to find t such that
Replace N(t) with and U(t) with
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Example 6-4a

22. Property of Inequality for Logarithms
Product Property of Logarithms
Inverse Property of Exponents and Logarithms
Subtract ln 278 and t from each side.
Divide each side by –0.017.
Use a calculator.
Answer: After 46 years or in 2046, Nigeria’s population will be greater than the population of the U.S.
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Example 6-4b

23. Answer: after 109 years or in the year 2109
Population As of 2000, Saudi Arabia had an estimated population of 20.7 million people and the United States had an estimated population of 278 million people. The growth of the populations of Saudi Arabia and the United States can be modeled by and , respectively. According to these models, when will Saudi Arabia’s population be more than the population of the United States?
Answer: after 109 years or in the year 2109
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Example 6-4c

24. Homework
See Syllabus 10.5
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