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126 Exercises 2.1 #R Chapter 2 1. f(x)=(!)* 3. f(x)= (0.4)" • EXPONENTIALS AND LOGARITHMS Practice Exercises 2.1 1. Predict the population of the United States for the year 2020 2. Suppose you put $8,000 in an account paying 9% interest assuming that it is modeled by the exponential function 281(1.02) in millions, where is the number of p() years after 2000. compounded annually. Find the balance in your account at the end of t years. Then, compute your balance for := 10 5. f(x) = ()' 7. f(x) = ()' 9. f(x)=()* population growth (and like compound interest), but in the opposite direction. Think of it this way: An initial amount of radon gas loses 16.5% every day, so that 83.5% - left. Starting with an initial amount of yo units, we have yo(0.835) at the end of the firs day. That amount becomes the starting amount for the second day, so by the end of the second day there is yo(0.835). At the end of the third day there is 83.5% of that amoun left-in other words, yo(0.835). Continuing this process, we see that the amount lef at the end of t days is yo(0.835). How many days will it take the original amount to reduce by half-that is, to go from yo to yo/27 Surprisingly, the answer to that question does not depend on the initia amount-in other words, it is the same for every yo. It is called the half-life of the substance. In order to find it, we need to know something about logarithms, which we will study in Section 2.3. In Exercises 1-4, use a calculator or spreadsheet to make a table of values of the function for x from -3 to 3 in steps of 0.5. Then plot the points and connect them to get a rough sketch of the graph 2. f(x)=()' 4. f(x) = 45 In Exercises 5-12, use the tables from the preceding prob lems but not a calculator or spreadsheet to make a table of values of each of the following functions for x from-3 to 3 in steps of 0.5. (Round the values to two decimal places.) Then plot the points and connect them to get a rough sketch of the graph 6. f(x)=()* 8. f(x) (0.25) 10. f(x)=(3) 12. f(x)=4* 11. f(x)=()* 13. The number 3 is irrational, and it has the infinite decimal expansion √3= 1.7320508.... Compute the sequence of numbers 2¹, 2¹7, 217, 212, and so forth and use it to estimate 23. Then use a calculator to find 2 and com- pare it with your estimate. Is the answer your calculator gives the precise value of 22 14. The number is famous in elementary geometry as the ratio of the circumference of a circle to its diameter. It i is an irrational number with infinite decimal expansion x=3.141592654.... Use the decimal expansion to cos struct a table of values for 2", where is a sequence of rational numbers tending toward . What do you infer from your table about the value of 27 Compare your estimate to the answer you get from a calculator In Exercises 15-22, assume that b is a positive number and that u and u are fixed numbers so that b=3 and b 4. Find a specific numerical value for each of the given expressions. 15. b 17. b 19. b 21. 1 16. b 18. 20, 2 22. 6- 23. Suppose that $500 is put in an account paying 7% interest compounded annually. (a) Write a formula giving the amount in the account a the end of years. (b) Determine how much there will be in the account a the end of 10 years. 24. Suppose that $2,000 is put in an account paying 4.5% in terest, compounded annually. (a) Write a formula giving the amount in the account a the end of years. (b) Determine how much there will be in the account a the end of 3 years.

2.1 Exponential Functions 25. Suppose you invest a certain amount of money at 6% in- terest, compounded annually. By what factor will your investment increase after 7 years? after 12 years? 26. How much money should be put in an account paying 5% interest, compounded annually, in order to have $5,000 eight years from now? (Hint: Write the formula for the amount after eight years on an initial investment of P dol- lars, set it equal to 5,000, and solve for P.) 27. How much money should be put in an account paying 4% interest, compounded annually, in order to have $50,000 twenty years from now? 28. Which of the following will yield the greater amount: (a) putting $1,000 in an account paying 3% interest, com- pounded annually, and leaving it for 10 years, or (b) putting $1,000 in an account paying 6% interest, compounded an- nually, and leaving it for 5 years? 29. A certain bacteria culture grows exponentially. In 1 hour the population grows from 500,000 to 800,000. Write a formula expressing the population P as a function of the time / in hours. 30. A colony of fruit flies is growing exponentially. At the beginning there were 3,000 flies, and the end of 3 days there were 7,000 flies. (a) Write a formula expressing the population P as a func- tion of the time t in days. (b) Determine how many flies there will be at the end of 5 days. 31. A colony of ants is increasing at a rate of 11% a day. At the end of the fifth day there are 580 ants. How many were there at the beginning of the first day? at the beginning of the second day? 127 32. Suppose a certain quantity of radon gas decayed to 400 cubic centimeters over a period of 8 days. How much was there to begin with? (Hint: Refer to Eq. (7).) 33. A certain radioactive substance decays in accordance with the formula y yo(0.88), where yo is the initial amount and y is the amount after years. What percentage of the original amount will be left after 5 years? 34. A certain radioactive substance decays in accordance with the formula y = y(0.93), where yo is the initial amount and r is the time in days. If there are 200 grams of the substance at the end of the third day, how many will there be at the end of the fifth day? 35. Plutonium-230 is a radioactive substance, whose decay equation has the form y=yob, where y is the amount at the end of r years. An initial amount of 50 milligrams of plutonium-230 decays to approximately 30 milligrams in 10 years. (a) Use the given data to find the constants yo and b. Round your answer to two decimal places. (You can use a calculator for b. but you shouldn't need one to find yo.) (b) How much of the initial 50 milligrams will be left at the end of 20 years? (c) Suppose the initial amount is 80 milligrams. How much will be left at the end of 15 years? 36. Suppose $1,000 is invested at an annual interest rate of 4.5%. Create a table of values and a chart of A(r), the amount after years, for r from 1 to 25. 37. Suppose $5,000 is invested at an annual interest rate of 7%. Create a table of values and a chart of the amount after 1 years for r from 1 to 30. 38. Suppose $1,000 is invested at an annual interest rate r. Create a table of values and a chart showing the amount at the end of 10 years for values of r from 0.03 to 0.05 in steps of 0.0025. 39. The world population in the year 2000 was approximately 6.16 billion. Assume it is growing exponentially at a rate of 1.0145% a year. Make a table of the projected popula- tion for each of the years from 2001 to 2020.

126 Practice Exercises 2.1 Chapter 2- EXPONENTIALS AND LOGARITHMS population growth (and like compound interest), but in the opposite direction. Thi left. Starting with an initial amount of yo units, we have yo(0.855) at the end of the of it this way: An initial amount of radon gas loses 16.5% every day, so that 83.5% day. That amount becomes the starting amount for the second day, so by the end of se second day there is yo(0.835). At the end of the third day there is 83.5% of that amo left-in other words. (0.835). Continuing this process, we see that the amount at the end of t days is (0.835). How many days will it take the original amount to reduce by half-that is, from yo to yo/2? Surprisingly, the answer to that question does not depend on the i amount-in other words, it is the same for every yo It is called the half-life of substance. In order to find it, we need to know something about logarithms, which will study in Section 2.3. 1. Predict the population of the United States for the year 2020 2 Suppose you put $8,000 in an account paying t compounded annually. Find the balance in your acco assuming that it is modeled by the exponential function pit) 281(1.02) in millions, where r is the number of years after 2000. the end off years. Then, compute your balance for a Exercises 2.1 In Exercises 1-4, use a calculator or spreadsheet to make a table of values of the function for x from 3 to 3 in steps of 0.5 Then plot the points and connect them to get a rough sketch of the graph 2. f(x)-()* 1. 10) = (3) 3. f(x) (0.4) In Exercises 5-12, use the tables from the preceding prob lems but not a calculator or spreadsheet to make a table of values of each of the following functions for x from-3 to 3 in steps of 0.5. (Round the values to two decimal places) Then plot the points and connect them so get a rough sketch of the graph 5. f(x)=()* 7. f(x)=(1) 9. 700)=()" 11./00=()" 6. /00=(1) 8. f(x) = (0.25) 10. f(x)=()** 12. f(x)=47 13. The number 3 is irrational, and it has the infinite decimal expansion √3-17320508.... Compute the sequence of numbers 2¹, 21, 21, 212, and so forth and use it to estimate 2 Then use a calculator to find 2 and com pare it with your estimate. Is the answer your calculator gives the precise value of 29 14. The number is famous in elementary geometry as the ratio of the circumference of a circle to its diameter. It is an irrational number with infinite decimal expan 3.141592654.... Use the decimal expansion to c struct a table of values for 2. where r is a sequenz rational numbers tending toward. What do you from your table about the value of 2"? Compare you estimate to the answer you get from a calculator 15. 17. 19. In Exercises 15-22, assume that b is a positive numbe and that a and are fited numbers so that bac = 4 Find a specific numerical value for each of given expressions 16. b 18. 8 20. 21. 22. 23. Suppose that $500 is put in an account paying 75 compounded annually. (a) Write a formula giving the amount in the acco the end of r years. (b) Determine how much there will be in the accou the end of 10 years 24. Suppose that $2.000 is put in an account paying 4.5 terest, compounded annually. (a) Write a formula giving the amount in the acc the end of r years. (b) Determine how much there will be in the acco the end of 3 years.

2.1 Exponential Functions 25. Suppose you invest a certain amount of money at 6% in- terest, compounded annually. By what factor will your investiment increase after 7 years? after 12 years? 26. How much money should be put in an account paying 5% 33. A certain radioactive substance decays in accordance with interest, compounded annually, in order to have $5,000 the formula y yo(0.88), where yo is the initial amount eight years from now? (Hint: Write the formula for the and y is the amount after r years. What percentage of the amount after eight years on an initial investment of P dol- original amount will be left after 5 years? lars, set it equal to 5,000, and solve for P.) 27. How much money should be put in an account paying 4% interest, compounded annually, in order to have $50,000 twenty years from now? 28. Which of the following will yield the greater amount: (a) putting $1,000 in an account paying 3% interest.com- pounded annually, and leaving it for 10 years, or (b) putting $1,000 in an account paying 6% interest, compounded an- nually, and leaving it for 5 years? 29. A certain bacteria culture grows exponentially. In 1 hour the population grows from 500,000 to 800,000. Write a formula expressing the population P as a function of the time t in hours. 30. A colony of fruit flies is growing exponentially. At the beginning there were 3,000 flies, and the end of 3 days there were 7,000 flies. (a) Write a formula expressing the population P as a func- tion of the time r in days. (b) Determine how many flies there will be at the end of 5 days. 127 32. Suppose a certain quantity of radon gas decayed to 400 cubic centimeters over a period of 8 days. How much was there to begin with? (Hint: Refer to Eq. (7).) 31. A colony of ants is increasing at a rate of 11% a day. At the end of the fifth day there are 580 ants. How many were there at the beginning of the first day? at the beginning of the second day? Solutions to practice exercises 2.1 1. To predict the U.S. population for 2020, we simply compute p(20). We have p()=281(1.02) 417.551 million. 34. A certain radioactive substance decays in accordance with the formula y yo(0.93), where yo is the initial amount and r is the time in days. If there are 200 grams of the substance at the end of the third day, how many will there be at the end of the fifth day? 35. Plutonium-230 is a radioactive substance, whose decay equation has the form y yub, where y is the amount at the end of t years. An initial amount of 50 milligrams of plutonium-230 decays to approximately 30 milligrams 10 years. (a) Use the given data to find the constants yo and b Round your answer to two decimal places. (You can use) a calculator for b, but you shouldn't need one to find y (b) How much of the initial 50 milligrams will be left at the end of 20 years? (c) Suppose the initial amount is 80 milligrams. How much will be left at the end of 15 years? 36. Suppose $1,000 is invested at an annual interest rate of 4.5%. Create a table of values and a chart of A(r), the amount after years, for r from 1 to 25. 37. Suppose $5,000 is invested at an annual interest rate of 7%. Create a table of values and a chart of the amount after years fort from 1 to 30. 38. Suppose $1,000 is invested at an annual interest rate r. Create a table of values and a chart showing the amount at the end of 10 years for values of r from 0.03 to 0.05 in steps of 0.0025. 39. The world population in the year 2000 was approximately 6.16 billion. Assume it is growing exponentially at a rate. of 1.0145% a year. Make a table of the projected popula tion for each of the years from 2001 to 2020. 2 2. If y(r) denotes the balance r years later, then using for- mula (3), we have y(t) = 8,000(1+0.09), or y(t) = 8.000(1.09), and 10 years later it will be equal to y(10) 8,000(1.09) 18.938.91.