Modeling with Exponential Functions

Transcription

1 CHAPTER Modeling with Exponential Functions A nautilus is a sea creature that lives in a shell. The cross-section of a nautilus s shell, with its spiral of ever-smaller chambers, is a natural example of an infinite geometric series, where each succeeding chamber is smaller than its neighbor by a constant proportion. A nautilus shell is particularly pleasing to the eye because the proportion of its chambers is the golden ratio, or approximately to 1. You will learn the properties of arithmetic and geometric series and how they are found in nature..1 Growth, Decay, and Interest Exponential Models p Too Much Homework! Geometric Sequences, Series, and Partial Sums p Sequences, Series, and Exponentials Geometric Sequences and Series as Exponential Functions p People, Tea, and Carbon Dioxide Modeling Using Exponential Functions p. 37 Chapter Modeling with Exponential Functions 375

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3 .1 Growth, Decay, and Interest Exponential Models Objectives In this lesson, you will: Write exponential functions as models of problem situations. Use exponential models to solve problems. Key Terms compound interest half-life decay Problem 1 How Many Ancestors? Everyone has two parents, four grandparents, and eight great-grandparents. Each set of ancestors represents a generation. 1. Your great-great-great-great-great-great-grandparents represent eight generations ago. How many ancestors do you have eight generations ago? 2. Write an exponential function to represent the number of ancestors that you have n generations ago. 3. Use this function to calculate the number of ancestors you have 12 generations and 20 generations ago. 4. How many generations ago do you have 33,554,432, or 2 25, ancestors? Lesson.1 Exponential Models 377

4 5. How many generations ago do you have 17,17,86,184, or 234, ancestors? 6. Each generation represents approximately 30 years of time. Ancestors that are 25 and 34 generations ago would have been alive how many years ago? 7. Are the answers in Questions 4 and 5 possible? Explain. Problem 2 Population Growth In 1, the world s population was about 6 billion people. The world s population is increasing at the rate of about 1.3% per year. 1. Write an exponential function to model the world s population over time. 2. Graph this function on the grid. 378 Chapter Modeling with Exponential Functions

5 3. Using this function, what was the world s population in the year 2000? 2005? 4. Use the graph to estimate when the world s population will reach 7 billion people. 5. Use a graphing calculator to determine when the world s population will reach 7 billion people. 6. When will the world s population reach 8 billion people? 7. According to this model, when will the world s population be double what it was in 1? 8. According to this model, when will the world s population be triple what it was in 1?. In 2000, the population of Kuwait was approximately 1.7 million people. Kuwait s population is decreasing at a rate of 4.5% annually. Write an exponential function that models the population of Kuwait over time. 10. When did Kuwait s population decrease to 1.3 million? 11. When will Kuwait s population be 850,000? Lesson.1 Exponential Models 37

6 Problem 3 Compound Interest Compound interest occurs when interest is added to the principal each period and the interest for the next period is calculated on this new principal. The formula for the principal calculated using compound interest after t years is ( P P0 1 r n ) nt where P is the current principal, P0 is the original principal, r is the annual rate of interest as a decimal, and n is the number of times per year that the interest is compounded. 1. Your friend was given the formula P 1000(1.0044)12t to calculate the return on his investment but he doesn t know what it means. Explain the formula for your friend. 2. Use a graphing calculator to determine how long it will take for your friend s investment to double. 4. Write an equation to determine how long it would take to double an investment of $1000 deposited in a Certificate of Deposit earning 4.5% annually, compounded monthly. Use a graphing calculator to solve this equation. 5. At 4.5% annual interest, compounded monthly, how long would it take an investment of $5000 to double? 6. How does the amount invested affect the doubling time? Explain. 380 Chapter Modeling with Exponential Functions 3. How long would it take for his investment to be worth $3500?

7 Problem 4 Half-Life Radioactive isotopes decay, or lose energy, at a fixed rate. The actual rate varies from one isotope to another. The term half-life refers to the time it takes for half of the atoms in a sample of a radioactive isotope to decay. Some radioactive isotopes have a half-life measured in billions of years, while others have a half-life of just a few seconds. For instance, strontium-0 is a radioactive isotope with a half-life of about 30 years. So, a 500-gram sample of strontium-0 will decay to 250 grams in 30 years. In 60 years, or two half-lives, 125 grams of the sample will remain. In 0 years, or three half-lives, 62.5 grams of the sample will remain. The formula for radioactive decay is t N N 0 (0.5) h where N is the amount of radioactive isotope left, N 0 is the original amount of radioactive isotope, t is the time that has passed, and h is the half-life. The half-life and the time are measured with the same units, such as seconds, days, weeks, months, or years. 1. A sample of strontium-0 contains 500 grams. Write a formula for the amount of strontium-0 remaining after t years. 2. Determine the amount of strontium-0 that remains after a. 60 years b. 75 years c. 150 years d. 63 years Lesson.1 Exponential Models 381

8 3. Use a graphing calculator to determine how long it will take for the sample to be a. 100 grams b. 5 grams c. 55 grams 4. Carbon-14 is a radioactive isotope used to date plants and animals. It has a half-life of about 5730 years. A plant or animal that is alive has a constant amount of carbon-14, but when the plant or animal dies, the carbon-14 begins to decay. The amount of carbon-14 remaining can be used to determine the time since the plant or animal died. Write a formula for the percent of carbon-14 remaining in a plant t years after it died. Use 100 as the initial amount present. 5. Calculate the percent of carbon-14 remaining after a years b years c. 10,000 years 382 Chapter Modeling with Exponential Functions

9 d. 500,000 years e. 1,000,000 years Problem 5 Disease Pandemic One of the worst outbreaks of the flu occurred in 118. Within two years, one fifth of the world s population was infected. The flu was most deadly for people ages 20 to 40, which was unusual since influenza usually affects the elderly and young children. The outbreak was estimated to have infected 28% of all Americans and killed approximately 675,000 Americans. How could something like this happen? Suppose that ten people in your school have contracted a strain of the flu and that each day they infect three other people. 1. Write a function that models the number of people infected each day after t days. 2. How many more people would be infected after a. one day? b. one week? c. one 30-day month? Lesson.1 Exponential Models 383

10 3. How many days would it take for: a. The entire school of 1200 people to become infected? b. The entire town of 12,000 people to become infected? c. The state of 5 million people to become infected? d. The entire US population of 300 million people to become infected? e. The entire world population of 6.5 billion people to become infected? 4. Given the theoretical results using this model, it would seem that any flu infection could quickly spread throughout the world. Why haven t we had a pandemic like this since 118? Be prepared to share your work with another pair, group, or the entire class. 384 Chapter Modeling with Exponential Functions

11 .2 Too Much Homework! Geometric Sequences, Series, and Partial Sums Objectives In this lesson, you will: List the terms of geometric sequences. Define geometric sequences using explicit formulas. Define geometric sequences using recursive formulas. List the terms of geometric series. Determine the sums of geometric series. Key Terms geometric sequence common ratio geometric series Problem 1 Some students are complaining about the amount of homework. So, your math teacher proposes a new method for assigning homework. His new homework proposal is to assign one problem on the first day, two on the second day, four on the third day, and so on. 1. Write a sequence to show the number of homework problems given to the class on the first through the fifth days. 2. Is this sequence an arithmetic sequence? Explain. 3. Write a recursive formula to calculate the number of homework problems given to the class on the nth day. 4. You previously wrote recursive formulas for arithmetic sequences. How is the formula from Question 3 similar? How is it different? Lesson.2 Geometric Sequences, Series, and Partial Sums 385

12 5. Is it more appropriate to model the number of homework problems using a sequence or a function? Explain. The sequence you wrote to represent your teacher s proposal is one example of a geometric sequence. A geometric sequence is a sequence that can be defined as g n r g n 1 where g n is the nth term in the sequence, g n 1 is the term before the nth term in the sequence, and r is the common ratio. The common ratio is the ratio between any two consecutive terms. 6. Complete the table and generate an explicit formula to calculate the number of problems received on the nth day. Day Homework Problems Received Calculation Using Multiplication Calculation Using Exponents n 386 Chapter Modeling with Exponential Functions

13 7. Suppose the class received 10 homework problems on the first day, and then the number of problems received doubled each day. Complete the table and generate an explicit formula to calculate the number of problems received on the nth day. Day Homework Problems Received Calculation Using Multiplication Calculation Using Exponents n 8. Examine the formulas you wrote in the last row of each table. How are the formulas similar? How are they different?. How is the common ratio indicated in the explicit formula of a geometric sequence? 10. How is the first term indicated in the explicit formula of a geometric sequence? 11. Write an explicit formula to calculate the nth term of each geometric sequence. a. The first term is 5 and the common ratio is 3. b. 40, 60, 0, 135, 202.5, Lesson.2 Geometric Sequences, Series, and Partial Sums 387

14 c. 3, 6, 12, 24, 48, d. The first term is g 1 and the common ratio is r. 12. Write a formula for the general geometric sequence, a. using a recursive formula; b. using an explicit formula. Problem 2 A geometric series is a series where each term is a member of a geometric sequence. As with arithmetic series, geometric series can be written using sigma notation. For example, consider the geometric series The first term, g 1, is equal to 120. The last term, g 4, is equal to The common ratio is. 2 An explicit formula for the nth term of the series is. The geometric series can be written using sigma notation as. S n For each series, identify g 1, r, g n, and S n. Then write each series using sigma notation. a ( 1 2 ) n 1 4 i ( 1 2 ) n 1 g 1 r g 4 S Chapter Modeling with Exponential Functions

15 b g 1 r g 5 S 5 c g 1 r g 5 S 5 d ,500 62, ,500 g 1 r g 6 S 6 2. Before deriving a formula for the sum of the first n terms of a geometric series, let s first derive the formula for the sum of the six terms of a geometric series, S 6 g 1 g 2 g 3 g 4 g 5 g 6. a. Rewrite each term as the product of the first term and the common ratio to a power. S 6 g 1 g 2 g 3 g 4 g 5 g 6 b. Factor the common factor of all the terms. S 6 r 1 c. Multiply the factored expression by and simplify. r 1 S 6 Lesson.2 Geometric Sequences, Series, and Partial Sums 38

16 3. Use a similar method to derive a formula for the sum of the first n terms of a geometric series. S n g 1 g 2 g 3... g n 1 g n 4. Use the formula from Question 3 to verify your answers in Question 1. a b Chapter Modeling with Exponential Functions

17 c d ,500 62, ,500 Be prepared to share your work with another pair, group, or the entire class. Lesson.2 Geometric Sequences, Series, and Partial Sums 31

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19 .3 Sequences, Series, and Exponentials Geometric Sequences and Series as Exponential Functions Objectives In this lesson, you will: Graph geometric sequences. Define geometric sequences as exponential functions. Graph geometric series. Define geometric series as exponential functions. Problem 1 1. For each geometric sequence, identify the first term g 1, the tenth term g 10, and the common ratio, r. Then create a scatter plot of the first ten terms with the number of the term as the independent variable. a. 1, , 3.375, b. g 1 3 g n g n 1 c. h 1 16 h n 0.5h n 1 Lesson.3 Geometric Sequences and Series as Exponential Functions 33

20 2. Graph each exponential function on the same grid. a. f(x) 1.5 x 1 b. g(x) 3(1.2) x 1 c. h(x) 16(0.5) x 1 3. How do the scatter plots of the geometric sequences differ from the graphs of the exponential functions? 4. What is the domain of each exponential function? 5. Explain how to change this domain so that the graph of the function is the same as the scatter plot of the corresponding sequence. 6. What conclusion can you make about writing a geometric sequence as an exponential function? 7. Write each geometric sequence as an exponential function. Be sure to define the appropriate domain. a. 1, 4, 16, 64,... b. g g n g n 1 c. h h n 2h n 1 8. Write each exponential function as a geometric sequence. a. f(x) 3 x 1 Domain is all counting numbers. b. f(x) 1000(0.8) x 1 Domain is all counting numbers. 34 Chapter Modeling with Exponential Functions

21 c. f(x) 10(1.2) x 1 Domain is all counting numbers. Problem 2 1. For each geometric sequence, calculate the first five partial sums. Then create a scatter plot of the first five partial sums with the number of the partial sum as the independent variable. a. 1, 1.5, 2.25, 3.375, b. g 1 3 g n g n 1 c. h 1 16 h n 0.5h n 1 2. Graph each exponential function on the same grid. a. b. f(x) 1.5x g(x) 3(1.2x 1) 0.2 c. h(x) 16(0.5x 1) 0.5 Lesson.3 Geometric Sequences and Series as Exponential Functions 35

22 3. How do the scatter plots of the partial sums differ from the graphs of the exponential functions? 4. What is the domain of each exponential function? 5. Explain how to change this domain so that the graph of the function is the same as the scatter plot of the corresponding partial sums. 6. What conclusion can you make about writing the partial sums of geometric sequences as an exponential function? 7. For each geometric sequence, rewrite the partial sums as an exponential function. Be sure to define the appropriate domain. a. 1, 2, 4, 8, g 1 16 g n g n 1 b. h 1 1 h n 3h n 1 Be prepared to share your methods and solutions. 36 Chapter Modeling with Exponential Functions

23 .4 People, Tea, and Carbon Dioxide Modeling Using Exponential Functions Objectives In this lesson, you will: Write exponential models from data sets. Use models to solve problems. Problem 1 US Population The table shows the population of the United States in millions of people at 20-year intervals from 1815 to 175. Year Population of the United States Lesson.4 Modeling Using Exponential Functions 37

24 1. Create a scatter plot of the data in the table with time since 1815 as the independent quantity and population as the dependent quantity. 2. Use a graphing calculator to determine the exponential regression equation and the value of the correlation coefficient, r. Graph the equation on the grid with the scatter plot. 3. Use the exponential regression equation to predict the population of the United States in The actual population of the United States in 10 was 258 million. Why does the model predict a different population? 5. Use the exponential regression equation to predict the population of the United States in What event in US history may explain why population increased more rapidly than the model predicts for years shortly after 170? 38 Chapter Modeling with Exponential Functions

25 Problem 2 Tea The table shows the temperature of a cup of tea over time. Time Minutes Temperature Degrees Fahrenheit Create a scatter plot of the data in the table with time as the independent quantity and temperature as the dependent quantity. Lesson.4 Modeling Using Exponential Functions 3

26 2. Use a graphing calculator to determine the exponential regression equation and the value of the correlation coefficient, r. Graph the equation on the grid with the scatter plot. 3. Use the exponential regression equation to determine the temperature of the tea after 60 minutes. 4. Use the exponential regression equation to determine when the tea will reach room temperature of 72 F. 5. What is the domain and range of this problem situation? Explain. 400 Chapter Modeling with Exponential Functions

27 Problem 3 Carbon Dioxide One measure of climate change is the amount of carbon dioxide in the Atlantic Ocean. When the level of carbon dioxide in the atmosphere increases, the concentration of carbon dioxide in the ocean water also increases. The table shows the carbon dioxide concentration in the Atlantic Ocean in parts per million from 1750 to 175. Year Carbon Dioxide Concentration Create a scatter plot of the data in the table with time since 1750 as the independent quantity and carbon dioxide concentration as the dependent quantity. Lesson.4 Modeling Using Exponential Functions 401

28 2. Use a graphing calculator to determine the exponential regression equation and the value of the correlation coefficient, r. Graph the equation on the grid with the scatter plot. 3. Use the exponential regression equation to predict the concentration of carbon dioxide in the Atlantic Ocean in the year Use the exponential regression equation to predict when the concentration of carbon dioxide in the Atlantic Ocean was 250 parts per million. 5. Use the exponential regression equation to predict when the concentration of carbon dioxide in the Atlantic Ocean was 100 parts per million. 6. According to the model, the concentration of carbon dioxide has been increasing exponentially over the past 250 years. What factors could have contributed to this behavior? Be prepared to share your methods and solutions. 402 Chapter Modeling with Exponential Functions