Chapter 10.6 Exponentials Growth and Decay Standard & Honors Algebra II Mr. Gilbert Chapter 10.6 Exponentials Growth and Decay Standard & Honors 11/24/2018

Agenda Warm up Work book (participation grade) Lesson Homework 11/24/2018

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Homework Review 11/24/2018

Communicate Effectively Decay Equation Growth Equation Natural base (e) Common base (10) 11/24/2018

(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) 11/24/2018 Lesson 6 Contents

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 11/24/2018 Example 6-1a

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. 11/24/2018 Example 6-1b

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 0.11 and t with 20. Ten percent of 130 is 13, so the answer seems reasonable. 11/24/2018 Example 6-1c

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: 13.8 hours 11/24/2018 Example 6-1d

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. 11/24/2018 Example 6-2a

Use a calculator. Answer: The constant k for Sodium-22 is 0.2666. Thus, the equation for the decay of Sodium-22 is where t is given in years. 11/24/2018 Example 6-2b

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 11/24/2018 Example 6-2c

Inverse Property for Exponents and Logarithms Divide each side by –0.2666. Use a calculator. Answer: It was formed about 9 years ago. 11/24/2018 Example 6-2d

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 11/24/2018 Example 6-2e

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 11/24/2018 Example 6-3a

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 11/24/2018 Example 6-3b

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

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 11/24/2018 Example 6-3d

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 11/24/2018 Example 6-4a

Property of Inequality for Logarithms Product Property of Logarithms Inverse Property of Exponents and Logarithms Subtract ln 278 and 0.026t 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. 11/24/2018 Example 6-4b

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 11/24/2018 Example 6-4c

Homework See Syllabus 10.5 11/24/2018