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21. Growth Constant for a Bacteria Culture A bacteria culture that exhibits exponential growth quadruples in size in 2 days. (a) Find the growth constant if time is measured in days. (b) If the initial size of the bacteria culture was 20,000, what is its size after just 12 hours?

21. Growth Constant for a Bacteria Culture A bacteria culture that exhibits exponential growth quadruples in size in 2 days. (a) Find the growth constant if time is measured in days. (b) If the initial size of the bacteria culture was 20,000, what is its size after just 12 hours?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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262 CHAPTER 5 Applications of the Exponential and Natural Logarithm Functions EXERCISES 5.1 In Exercises 1-10, determine the growth constant k, then find all solutions of the given differential equation. 1. y' = y 2. y' = 4y 3. y' = 1.7y 5. y'-/=0 7. 2y'--0 9. 1-4 11. y'= 3y, y(0) = 1 13. y'= 2y, y(0) = 2 4. y'= 6. y' 6y=0 8. y 1.6y' 10. 5y' 6y=0 In Exercises 11-18, solve the given differential equation with initial condition. 12. y' = 4y, y(0) = 0 14. y'= y, y(0) = 4 16. y' - 15. y' 6y = 0, y(0) = 5 17. 6y' y, y(0) = 12 18. Sy 3y', y(0) = 7 19. Population with Exponential Growth Let P(1) be the popula- tion (in millions) of a certain city years after 2015, and sup- pose that P(t) satisfies the differential equation P'(t) = .01P(t), P(0) = 2. = 0, y(0) = 6 (a) Find a formula for P(t). (b) What was the initial population, that is, the population in 2015? (c) Estimate the population in 2019. 20. Growth of a Colony of Fruit Flies A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(1) denote the number of fruit flies / days later, and let k= .08 denote the growth constant. (a) Write a differential equation and initial condition that model the growth of this colony. (b) Find a formula for P(t). (c) Estimate the size of the colony 5 days later. 21. Growth Constant for a Bacteria Culture A bacteria culture that exhibits exponential growth quadruples in size in 2 days. (a) Find the growth constant if time is measured in days. (b) If the initial size of the bacteria culture was 20,000, what is its size after just 12 hours? 22. Growth of a Bacteria Culture The initial size of a bacteria cul- ture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria. (a) Find the growth constant if time is measured in days. (b) How long will it take for the culture to double in size? 23. Using the Differential Equation Let P(t) be the population (in millions) of a certain city years after 2015, and suppose that P(1) satisfies the differential equation P'(t)= .03P(1), P(0) = 4. (a) Use the differential equation to determine how fast the population is growing when it reaches 5 million people. (b) Use the differential equation to determine the population size when it is growing at the rate of 400,000 people per year. (c) Find a formula for P(1). 24. Growth of Bacteria Approximately 10,000 bacteria are placed in a culture. Let P(1) be the number of bacteria present in the culture after t hours, and suppose that P(1) satisfies the dif- ferential equation P'(t) = .55P(t). (a) What is P(0)? (b) Find the formula for P(1). (c) How many bacteria are there after 5 hours? (d) What is the growth constant? (e) Use the differential equation to determine how fast the bacteria culture is growing when it reaches 100,000. (f) What is the size of the bacteria culture when it is growing at a rate of 34,000 bacteria per hour? 25. Growth of Cells Afterhours there are P(1) cells present in a culture, where P(t) = 50000.21 (a) How many cells were present initially? (b) Give a differential equation satisfied by P(t). (c) When will the initial number of cells double? (d) When will 20,000 cells be present? 26. Insect Population The size of a certain insect population is given by P(1) = 300e0.011, where t is measured in days. (a) How many insects were present initially? (b) Give a differential equation satisfied by P(t). (c) At what time will the initial population double? (d) At what time will the population equal 1200? 27. Population Growth Determine the growth constant of a popu- lation that is growing at a rate proportional to its size, where the population doubles in size every 40 days and time is mea- sured in days. 28. Time to Triple Determine the growth constant of a popula- tion that is growing at a rate proportional to its size, where the population triples in size every 10 years and time is measured in years. 29. Exponential Growth A population is growing exponentially with growth constant .05. In how many years will the current population triple? 30. Time to Double A population is growing exponentially with growth constant .04. In how many years will the current popu- lation double? 31. Exponential Growth The rate of growth of a certain cell cul- ture is proportional to its size. In 10 hours a population of 1 million cells grew to 9 million. How large will the cell culture be after 15 hours? 32. World's Population The world's population was 5.51 billion on January 1, 1993, and 5.88 billion on January 1, 1998. Assume that, at any time, the population grows at a rate proportional to the population at that time. In what year will the world's population reach 7 billion? 33. Population of Mexico City At the beginning of 1990, 20.2 mil- lion people lived in the metropolitan area of Mexico City, and the population was growing exponentially. The 1995 popula- tion was 23 million. (Part of the growth is due to immigra- tion.) If this trend continues, how large will the population be in the year 2010?
Transcribed Image Text:262 CHAPTER 5 Applications of the Exponential and Natural Logarithm Functions EXERCISES 5.1 In Exercises 1-10, determine the growth constant k, then find all solutions of the given differential equation. 1. y' = y 2. y' = 4y 3. y' = 1.7y 5. y'-/=0 7. 2y'--0 9. 1-4 11. y'= 3y, y(0) = 1 13. y'= 2y, y(0) = 2 4. y'= 6. y' 6y=0 8. y 1.6y' 10. 5y' 6y=0 In Exercises 11-18, solve the given differential equation with initial condition. 12. y' = 4y, y(0) = 0 14. y'= y, y(0) = 4 16. y' - 15. y' 6y = 0, y(0) = 5 17. 6y' y, y(0) = 12 18. Sy 3y', y(0) = 7 19. Population with Exponential Growth Let P(1) be the popula- tion (in millions) of a certain city years after 2015, and sup- pose that P(t) satisfies the differential equation P'(t) = .01P(t), P(0) = 2. = 0, y(0) = 6 (a) Find a formula for P(t). (b) What was the initial population, that is, the population in 2015? (c) Estimate the population in 2019. 20. Growth of a Colony of Fruit Flies A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(1) denote the number of fruit flies / days later, and let k= .08 denote the growth constant. (a) Write a differential equation and initial condition that model the growth of this colony. (b) Find a formula for P(t). (c) Estimate the size of the colony 5 days later. 21. Growth Constant for a Bacteria Culture A bacteria culture that exhibits exponential growth quadruples in size in 2 days. (a) Find the growth constant if time is measured in days. (b) If the initial size of the bacteria culture was 20,000, what is its size after just 12 hours? 22. Growth of a Bacteria Culture The initial size of a bacteria cul- ture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria. (a) Find the growth constant if time is measured in days. (b) How long will it take for the culture to double in size? 23. Using the Differential Equation Let P(t) be the population (in millions) of a certain city years after 2015, and suppose that P(1) satisfies the differential equation P'(t)= .03P(1), P(0) = 4. (a) Use the differential equation to determine how fast the population is growing when it reaches 5 million people. (b) Use the differential equation to determine the population size when it is growing at the rate of 400,000 people per year. (c) Find a formula for P(1). 24. Growth of Bacteria Approximately 10,000 bacteria are placed in a culture. Let P(1) be the number of bacteria present in the culture after t hours, and suppose that P(1) satisfies the dif- ferential equation P'(t) = .55P(t). (a) What is P(0)? (b) Find the formula for P(1). (c) How many bacteria are there after 5 hours? (d) What is the growth constant? (e) Use the differential equation to determine how fast the bacteria culture is growing when it reaches 100,000. (f) What is the size of the bacteria culture when it is growing at a rate of 34,000 bacteria per hour? 25. Growth of Cells Afterhours there are P(1) cells present in a culture, where P(t) = 50000.21 (a) How many cells were present initially? (b) Give a differential equation satisfied by P(t). (c) When will the initial number of cells double? (d) When will 20,000 cells be present? 26. Insect Population The size of a certain insect population is given by P(1) = 300e0.011, where t is measured in days. (a) How many insects were present initially? (b) Give a differential equation satisfied by P(t). (c) At what time will the initial population double? (d) At what time will the population equal 1200? 27. Population Growth Determine the growth constant of a popu- lation that is growing at a rate proportional to its size, where the population doubles in size every 40 days and time is mea- sured in days. 28. Time to Triple Determine the growth constant of a popula- tion that is growing at a rate proportional to its size, where the population triples in size every 10 years and time is measured in years. 29. Exponential Growth A population is growing exponentially with growth constant .05. In how many years will the current population triple? 30. Time to Double A population is growing exponentially with growth constant .04. In how many years will the current popu- lation double? 31. Exponential Growth The rate of growth of a certain cell cul- ture is proportional to its size. In 10 hours a population of 1 million cells grew to 9 million. How large will the cell culture be after 15 hours? 32. World's Population The world's population was 5.51 billion on January 1, 1993, and 5.88 billion on January 1, 1998. Assume that, at any time, the population grows at a rate proportional to the population at that time. In what year will the world's population reach 7 billion? 33. Population of Mexico City At the beginning of 1990, 20.2 mil- lion people lived in the metropolitan area of Mexico City, and the population was growing exponentially. The 1995 popula- tion was 23 million. (Part of the growth is due to immigra- tion.) If this trend continues, how large will the population be in the year 2010?
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