Exponential Growth and Decay Models #1

Transcription

1 Exponential Growth and Decay Models #1 Name: Please show all work, round all dollars to the nearest cent, and population to the nearest person 1. A major technology company, ExpoGrow, is growing incredibly fast. The latest prospectus (a report on the company) said that so far, the number of employees, y, could be found with the equation y = 3(4) x, where x represents the number of years since the company was founded. a. How many people founded the company? b. Describe the growth of the company in words. 2. As part of a major scandal, it was discovered that several statements in the prospectus for ExpoGrow in Question 1 were false. If the company actually had 5 founders and doubles in size each year, what equation should it have printed in its report? 3. Dusty won $125,000 on the Who Wants to be a Zillionaire? game show. He decides to place the money into an account that earns 6.25% interest compounded annually and plans not to use any of it until he retires. a. Write an expression that models this situation b. How much money will be in the account in 23 years? 4. A bank account starts with a balance of $100, and earns an annual interest rate of 4%. a. Write an expression that models b. How much money will be in the this situation. account after 10 years? 5. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after a. Write an expression that b. How many cell phone subscribers were models this situation. ther in Centerville in 1994?

2 6. Bacteria can multiply at an alarming rate, doubling every hour. Assume you start with one bacterium. a. Write an expression that models this situation. b. How many bacteria will be present after 12 hours? After one day? 7. Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. a. Write an expression that models this situation. b. How many players remain in the tournament after 5 rounds? 8. The population of Winnemucca, Nevada, can be modeled by p = 6191(1.045) t, where t is the number of years since a. What was the population in 1990? b. By what percent did the population increase each year? c. In what year will the population in Winnemucca reach 20,000? 9. You have inherited land that was purchased for $30,000 in Since then the value of the land has increased by approximately 5.1% per year. In what year did the land reach $90,000? 10. During normal breathing, about 12% of the air in the lungs is replaced after one breath. If the initial amount of air in the lungs is 500 ml, how much of the initial amount of air is present after 24 breaths? 11. An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person s system decreases by about 29%. How much ibuprofen is left after 6 hours?

3 12. In June, a 5th grade class started out with 10 mealworms for an experiment. The population of worms will double every month. b. How many mealworms will the class have for their experiment in... 3 months? 1 year? 45 days? 13. Your baby brother has an ear infection. The doctor said there are probably 50,000,000 bacteria in his left ear. The penicillin the doctor prescribed will kill 7% of the bacteria every 6 hours. b. How many bacteria will be in your brother s ear in... 1 day? 5 days? 3 hours? 14. The population of Succasunna, NJ was 11,171 in The estimated population growth is 2.1% every 4 years. b. What is the estimated population in the year ? 2001? this year? at the beginning of the next century? 15. In 2014 an estimated 4,324,000 people in this country were illiterate. With new incentives and funding, the country is hoping to cut that number by 11% every 3 years. b. How many people do you predict will be illiterate in the year ? 2095? 2155?

4 16. In Sub-Saharan Africa an estimated 28.5 million people are infected with AIDS. That number is estimated to grow at a rate of 0.4% every 2 years. b. How many people will be infected in... (round your answers to the nearest person) 3 years? 10 years? 1000 years? 17. Over the last several years there has been an increase in the amount of school age children carrying cell phones. Current estimates (in 2017) are that there are 1,237,000 students in the US with cell phones. The trend is likely to continue, with the amount of children with phones tripling every 10 years. b. How many students will have cell phones in... (round to the nearest person) 2022? 2100? in 2 years? 18. Coal was once a booming industry in central Pennsylvania. However, the industry has begun to decline. In the year 1950 about 600,000 were employed in the local coal mines. Since then the number of coal miners has declined by 5% every 6 years. b. Predict the number of coal workers employed in each year given

5 ANSWER KEY Please show all work, round all dollars to the nearest cent, and population to the nearest person 1. A major technology company, ExpoGrow, is growing incredibly fast. The latest prospectus (a report on the company) said that so far, the number of employees, y, could be found with the equation y = 3(4) x, where x represents the number of years since the company was founded. a. How many people founded the company? b. Describe the growth of the company in words. 3 QUADRUPLES EACH YEAR 2. As part of a major scandal, it was discovered that several statements in the prospectus for ExpoGrow in Question 1 were false. If the company actually had 5 founders and doubles in size each year, what equation should it have printed in its report? 3. Dusty won $125,000 on the Who Wants to be a Zillionaire? game show. He decides to place the money into an account that earns 6.25% interest compounded annually and plans not to use any of it until he retires. a. Write an expression that models this situation b. How much money will be in the account in 23 years? $504, A bank account starts with a balance of $100, and earns an annual interest rate of 4%. a. Write an expression that models b. How much money will be in the this situation. account after 10 years? $ In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after a. Write an expression that b. How many cell phone subscribers were models this situation. there in Centerville in 1994? 43,872

6 6. Bacteria can multiply at an alarming rate, doubling every hour. Assume you start with one bacterium. a. Write an expression that models this situation. b. How many bacteria will be present after 12 hours? After one day? 4,096 16,777, Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. a. Write an expression that models this situation. b. How many players remain in the tournament after 5 rounds? 4 8. The population of Winnemucca, Nevada, can be modeled by p = 6191(1.045) t, where t is the number of years since a. What was the population in 1990? 6191 b. By what percent did the population increase each year? 4.5% 9. You have inherited land that was purchased for $30,000 in Since then the value of the land has increased by approximately 5.1% per year. What is the approximate value of the land in the year 2017? $511, During normal breathing, about 12% of the air in the lungs is replaced after one breath. If the initial amount of air in the lungs is 500 ml, how much of the initial amount of air is present after 24 breaths? 23.3 ml 11. An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person s system decreases by about 29%. How much ibuprofen is left after 6 hours? mg

7 12. In June, a 5th grade class started out with 10 mealworms for an experiment. The population of worms will double every month. d. How many mealworms will the class have for their experiment in... 3 months? 1 year? 45 days? Your baby brother has an ear infection. The doctor said there are probably 50,000,000 bacteria in his left ear. The penicillin the doctor prescribed will kill 7% of the bacteria every 6 hours. d. How many bacteria will be in your brother s ear in... 1 day? 5 days? 3 hours? 374, , , The population of Succasunna, NJ was 11,171 in The estimated population growth is 2.1% every 4 years. d. What is the estimated population in the year ? 2001? this year? at the beginning of the next century? 12,394 11,229 12,202 18, In 2014 an estimated 4,324,000 people in this country are illiterate. With new incentives and funding, the country is hoping to cut that number by 11% every 3 years. d. How many people do you predict will be illiterate in the year ? 2095? 2155? 1,912, ,958 18,080

8 16. In Sub-Saharan Africa an estimated 28.5 million people are infected with AIDS. That number is estimated to grow at a rate of 0.4% every 2 years. d. How many people will be infected in... (round your answers to the nearest person) 3 years? 10 years? 1000 years? 28,671,171 29,074, ,749, Over the last several years there has been an increase in the amount of school age children carrying cell phones. Current estimates are that there are 1,237,000 students in the US with cell phones. The trend is likely to continue, with the amount of children with phones tripling every 10 years. c. Write an exponential function : d. How many students will have cell phones in... (round to the nearest person) 2022? 2100? in 2 years? 2,142,546 11,284,338,720 1,540, Coal was once a booming industry in central Pennsylvania. However, the industry has begun to decline. In the year 1950 about 600,000 were employed in the local coal mines. Since then the number of coal miners has declined by 5% every 6 years. d. Predict the number of coal workers employed in each year given , , ,399