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Common Core Algebra Unit 10: Exponential Functions 2/7/17Lesson 6: Growth and Decay ModelsObjective: SWBAT model and solve problems involving exponential growth and decay in word problems.Do Now: Answer the following questions below.1. Change each percent to a decimal.a) 20% b) 2.5% c) 0.1% d) 18.2%2. What is 12% of 1,200?Group Task: Read and solve the following word problems.(a) The radio station Z-100 is sponsoring a contest. The prize begins as a $1000 gift card to Roosevelt Field Mall. Once a day, the disc jockey announces a name, and the person has 15 minutes to call in and claim the prize. If the person does not call within the allotted time, the prize increases by 10% per day. How much will the gift card be worth if no one wins after 3 days?(b) Leo purchases a car for $26,499. The car depreciates (loses value) at a rate of 18% annually. What will Leo's car be worth after 3 years?Exponential Growth occurs when a quantity increases by the same rate, r, in each unit of time, t.Exponential Decay occurs when a quantity decreases by the same rate, r, in each unit of time, t.The value of the quantity at any given time can be calculated as a function of the rate and the original amount.Exponential Growth ModelExponential Decay Model4864104508500Let's look at the Group Task... which situation represents exponential growth? exponential decay?Exponential ModelExponential ModelWhat is the value of the prize money after 3 days have passed?What is the value of Leo’s car after 3 years?Model Example:Maria's parents invested $14,000 in a CD account earning 6% per year compounded annually. How much money will there be in the account after 10 years?Check for Understanding:In 2000, 2200 students attended Queens Metropolitan High School. The enrollment has since been declining 2% annually. If this trend continues, how many students will be enrolled in 2016?Compound InterestInterest is calculated once per period on the current amount borrowed or invested. Each period, the interest becomes a part of the principal.F = Future ValueP= Investment (Starting Amount)R= Rate (Percent as a Decimal)T= Time in yearsCheck for Understanding:Your 3 year investment of $20,000 received 5.2% interested compounded annually. What is your total return? Lesson Summary:If a relationship grows over time, it can be represented by an Exponential Growth model, __________________________, where 1 + r represents the _________________ _________________ between successive function values when t increases by 1.If a relationship decreases over time, it can be represented by an Exponential Decay model, __________________________, where 1 - r represents the _________________ _________________ between successive function values when t increases by 1.Partner Practice:1. A sculpture was valued at $1200 in the year 1990. Since then it has been appreciating at a rate of 8% per year.a) Write an exponential function to model this situation. ________________.b) Complete the table of values that shows the increase in value over time. Round to the nearest dollar.c) Sketch a graph of the function with the indicated window.d) How much is the sculpture worth now to the nearest dollar?2. Mr. Rogers purchased machinery for his farming operation for $175,000. It is expected to depreciate at a rate of 9% per year.a) Write an exponential function to model this situation. What will be the value of the piece of machinery in 10 years?b) Sketch a graph of the function with the indicated window.c) Approximately, how many years will it take for the combine to be worth $50,000?3. Ms. Arnold received a job as a teacher with a starting salary of $55,000. According to her contract, she will receive a 1.5% increase in her salary every year. Write an exponential function that can be used to find S, Ms. Arnold’s salary after t years. How many years will it take for Ms. Arnold to reach a minimum salary of $60,000?4. A fully inflated raft containing 4500 cubic inches of air loses 6.6% of its air every day.a) After 5 days, how much air remains in the raft? Round to the nearest cubic inch.b) How much air was lost?5. The current enrollment of the QMHS is expected to increase over the next five years. Each year the population is expected to increase by about 3.2% from the previous year. How many more students are expected to be enrolled in year 5 than in year 4 if the current enrollment is 850 students?6. In a particular state, the population of black bears has been decreasing at the rate of 0.75% per year. In 1990, it was estimated that there were 400 black bears in the state. If the population continues to decline at the same rate, what will the population be in 2017?7. Camilo purchased a rare coin from a dealer for $300. The value of the coin increases 5.5% each year. How many years will it take for the coin to increase in value by $100?8. Elise is buying a new car for $42,500. As time goes by, the value of the car will decrease. It’s worth can be estimated using the equation y = 42,500(0.91)x in which y represents the value of the car over x years.a) What is the depreciation rate, r, of this particular car? Express your answer as a percent.365760028575000b) Create a table of values that shows the car’s value over a period of 20 years.c) Using your table of values, create a graph over the interval 0 < x < 20. Think about the scale you will need to create on the y-axis in order to graph the function.Make sure to label axes.d) What will Elise’s car be worth after 15 years?Problem Set:1. Some banks charge a fee on savings accounts that are left inactive for an extended period of time. The equation represents the value, y, of one account that was left inactive for a period of x years. What is the y-intercept of this equation and what does it represent?1)0.98, the percent of money in the account initially2)0.98, the percent of money in the account after x years3)5000, the amount of money in the account initially4)5000, the amount of money in the account after x years2. The function represents the value , in dollars, of a comic book t years after its purchase. The yearly rate of appreciation of the comic book is1)17%2)1.7%3)1.017%4)0.017%3. The equation is being used to calculate the amount of money in a savings account. What does 1.02 represent in this equation?1)0.02% decay2)0.02% growth3)2% decay4)2% growth4. Milton has his money invested in a stock portfolio. The value, , of his portfolio can be modeled with the function , where x is the number of years since he made his investment. Which statement describes the rate of change of the value of his portfolio?1)It decreases 78% per year.2)It decreases 22% per year.3)It increases 78% per year.4)It increases 22% per year.5. Is the equation a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period?1)exponential growth and 12%2)exponential growth and 88%3)exponential decay and 12%4)exponential decay and 88%6. The number of carbon atoms in a fossil is given by the function , where x represents the number of years since being discovered. What is the percent of change each year? Explain how you arrived at your answer.7. The current population of a town is 10,000. If the population, P, increases by 20% each year, which equation could be used to find the population after t years?1)2)3)4)8. Robert invests $800 in an account at 1.8% interest compounded annually. He will make no deposits or withdrawals on this account for 3 years. Which formula could be used to find the balance, A, in the account after the 3 years?1)2)3)4)9. Krystal was given $3000 when she turned 2 years old. Her parents invested it at a 2% interest rate compounded annually. No deposits or withdrawals were made. Which expression can be used to determine how much money Krystal had in the account when she turned 18?1)2)3)4)10. Mr. Smith invested $2,500 in a savings account that earns 3% interest compounded annually. He made no additional deposits or withdrawals. Which expression can be used to determine the number of dollars in this account at the end of 4 years?1)2)3)4) 11. Sheba opened a retirement account with $36,500. Her account grew at a rate of 7% per year compounded annually. She made no deposits or withdrawals on the account. At the end of 20 years, what was the account worth, to the nearest dollar?1)$87,6002)$130,7863)$141,2434)$1,483,444,46312. Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually, what will be the value of Cassandra's dresser at the end of 3 years to the nearest dollar?1)$4152)$5903)$5964)$77013. The current student population of the Brentwood Student Center is 2,000. The enrollment at the center increases at a rate of 4% each year. To the nearest whole number, what will the student population be closest to in 3 years'?1)2,2402)2,2503)5,4884)6,24014. The population of Henderson City was 3,381,000 in 1994, and is growing at an annual rate of 1.8%. If this growth rate continues, what will the approximate population of Henderson City be in the year 2000?1)3,696,0002)3,763,0003)3,798,0004)3,831,00015. Adrianne invested $2000 in an account at a 3.5% interest rate compounded annually. She made no deposits or withdrawals on the account for 4 years. Determine, to the nearest dollar, the balance in the account after the 4 years.16. Kirsten invested $1000 in an account at an annual interest rate of 3%. She made no deposits or withdrawals on the account for 5 years. The interest was compounded annually. Find the balance in the account, to the nearest cent, at the end of 5 years.17. Dylan invested $600 in a savings account at a 1.6% annual interest rate. He made no deposits or withdrawals on the account for 2 years. The interest was compounded annually. Find, to the nearest cent, the balance in the account after 2 years.Name_____________________ Exit SlipCommon Core Algebra Unit 10: Exponential Functions 2/7/17Lesson 6: Growth and Decay ModelsObjective: SWBAT model and solve problems involving exponential growth and decay in word problems.The breakdown of a sample of a chemical compound is represented by the function , where represents the number of milligrams of the substance and t represents the time, in years. In the function , explain what 0.5 and 300 represent.Name_____________________ Exit SlipCommon Core Algebra Unit 10: Exponential Functions 2/7/17Lesson 6: Growth and Decay ModelsObjective: SWBAT model and solve problems involving exponential growth and decay in word problems.The breakdown of a sample of a chemical compound is represented by the function , where represents the number of milligrams of the substance and t represents the time, in years. In the function , explain what 0.5 and 300 represent. ................
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