1. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 1 Lesson 21 – Applications of Exponential Functions Precalculus

2. 2 BIG PICTURE Solve word problems involving exponential equations, using both algebraic & graphic approaches 11/23/2015 Precalculus - Lesson 21 - Exponential Models

3. (A) REVIEW: Exponential Functions The features of the parent exponential function y = a x (where a > 1) are as follows: The features of the parent exponential function y = a -x (where a > 1) are as follows: 11/23/2015 Precalculus - Lesson 21 - Exponential Models 3

4. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 4 (B) Exponential Modeling - Growth In general, however, the algebraic model for exponential growth is y = c(a) x where a is referred to as the growth ratio (provided that a > 1) and c is the initial amount present and x is the number of increases given the growth rate conditions. All equations in this section are also written in the form y = c(1 + r) x where c is a constant, r is a positive rate of change and 1 + r > 1, and x is the number of increases given the growth rate conditions.

5. Modeling with Exponential Fcns  Ratios & Rates Data set: Determine the ratio between successive y values: (y 2 /y 1, y 3 /y 2, y 4 /y 3, etc.)  y = ca x Determine the percent change between successive y values: (y 2 -y 1 )/y 1, (y 3 -y 2 )/y 2, (y 4 -y 3 )/y 3, etc  y = c(1 + r) x 11/23/2015 Precalculus - Lesson 21 - Exponential Models 5 X0123456 y5075112.5168.75253.13379.69569.53

6. (B) Exponential Modeling - Growth Populations can also grow exponentially according to the formula P(n) = P o (1.0125) n. If a population of 4,000,000 people grows according to this formula, determine: 1. what is the average annual rate of increase of the population 2. the population after 5 years 3. the population after 12.25 years 4. when will the population be 6,500,000 5. what is the MONTHLY rate of increase of the population? 11/23/2015 Precalculus - Lesson 21 - Exponential Models 6

7. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 7 (C) Exponential Modeling - Decay In general, however, the algebraic model for exponential decay is y = c(a) x where a is referred to as the decay ratio (provided that 0 < a < 1) and c is the initial amount present and x is the number of decreases given the decay rate conditions. All exponential equations are also written in the form y = c(1 + r) x where c is a constant, r is a negative rate of change and (1 + r) < 1, and x is the number of increases given the decay rate conditions.

8. (C) Exponential Modeling - Decay The value of a car depreciates according to the exponential equation V(t) = 25,000(0.8) t, where t is time measured in years since the car’s purchase. Determine: 1. the car’s value after 5 years 2. the car’s value after 7.5 years 3. when will the car’s value be $8,000 4. what is the average annual rate of decrease of the car’s value? 11/23/2015 Precalculus - Lesson 21 - Exponential Models 8

9. (D) Working with Exponential Models Bacteria grow exponentially according to the equation N(t) = N o 2 (t/d) where N(t) is the amount after a certain time period, N o is the initial amount, t is the time and d is the doubling period A bacterial strain doubles every 30 minutes. If the starting population was 4,000, determine 1. the number of bacteria after 85 minutes 2. the number of bacteria after 5 hours 3. when will the number of bacteria be 30,000 4. what is the average hourly rate of increase of the # of bacteria? 4. what is the average daily rate of increase of the # of bacteria? 11/23/2015 Precalculus - Lesson 21 - Exponential Models 9

10. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 10 (D) Working with Exponential Models Ex. 1 A bacterial strain doubles every 30 minutes. If there are 1,000 bacteria initially, how many are present after 6 hours? Ex 2. The number of bacteria in a culture doubles every 2 hours. The population after 5 hours is 32,000. How many bacteria were there initially?

11. 11 (D) Working with Exponential Models Ex 3. The number of bacteria doubles every hour. The initial population is 3600.  (a) Determine the bacterial population in 2 days.  (b) When will the bacterial population be 1,000,000,000? Ex 4. In a biology experiment, the number of cells present in 2 hours is 1,000. Eight hours later, the number of cells was estimated to be 250,000.  (a) Determine the doubling period.  (b) If we need 2,000,000 cells in order to start with the next experiment, how long do we need to wait? 11/23/2015 Precalculus - Lesson 21 - Exponential Models

12. 12 (D) Working with Exponential Models Ex 5. Based on data from the past 50 years, world population growth can be modeled by the exponential equation P = 2.5(1.0185) x, where P is population in billions and x is # of years since 1950.  (a) Explain how we know from the problem that the annual growth rate is 1.85%  (b) Estimate the world population in 2015  (c) When will the world population be double that of 2010 11/23/2015 Precalculus - Lesson 21 - Exponential Models

13. (D) Working with Exponential Models Ex 6. Radioactive chemicals decay over time according to the formula N(t) = N o 2 (-t/h) where N(t) is the amount after a certain time period, N o is the initial amount, t is the time and h is the halving time  which we can rewrite as N(t) = N o (1/2) (t/h) and the ratio of t/h is the number of halving periods. 320 mg of iodine-131 is stored in a lab but it half life is 60 days. Determine: 1. the amount of I-131 left after 10 days 2. the amount of I-131 left after 180 days 3. when will the amount of I-131 left be 100 mg? 4. what is the average daily rate of decrease of I-131? 4. what is the average monthly rate of decrease of I-131? 11/23/2015 Precalculus - Lesson 21 - Exponential Models 13

14. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 14 (D) Working with Exponential Models Ex 7. Health officials found traces of Radium F beneath the science labs at CAC. After 69 d, they noticed that a certain amount of the substance had decayed to 1/√2 of its original mass. Determine the half-life of Radium F

15. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 15 (D) Working with Exponential Models Investments grow exponentially as well according to the formula A = P o (1 + i) n. If you invest $500 into an investment paying 7% interest compounded annually, what would be the total value of the investment after 5 years? Ex 8. You invest $5000 in a stock that grows at a rate of 12% per annum compounded quarterly. The value of the stock is given by the equation V = 5000(1 + 0.12/4) 4x, or V = 5000(1.03) 4x where x is measured in years.  (a) Find the value of the stock in 6 years.  (b) Find when the stock value is $14,000

16. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 16 (D) Working with Exponential Models ex 9. The population of a small town was 35,000 in 1980 and in 1990, it was 57,010. Create an algebraic model for the towns population growth. Check your model using the fact that the population was 72800 in 1995. What will the population be in 2010? ex 10. Three years ago there were 2500 fish in Loon Lake. Due to acid rain, there are now 1945 fish in the lake. Find the population 5 years from now, assuming exponential decay.

17. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 17 (D) Working with Exponential Models ex 11. The value of a car depreciates by about 20% per year. Find the relative value of the car 6 years after it was purchased. Ex 12. When tap water is filtered through a layer of charcoal and other purifying agents, 30% of the impurities are removed. When the water is filtered through a second layer, again 30% of the remaining impurities are removed. How many layers are required to ensure that 97.5% of the impurities are removed from the tap water?

18. 18 (E) Working with A = Pe rt So our formula for situations featuring continuous change becomes A = Pe rt In the formula, if r > 0, we have exponential growth and if r < 0, we have exponential decay P represents an initial amount 11/23/2015 Precalculus - Lesson 21 - Exponential Models

19. 19 (E) Working with A = Pe rt Ex 1. I invest $10,000 in a funding yielding 12% p.a. compounded continuously.  (a) Find the value of the investment after 5 years.  (b) How long does it take for the investment to triple in value? Ex 2. The population of the USA can be modeled by the eqn P(t) = 227e 0.0093t, where P is population in millions and t is time in years since 1980  (a) What is the annual growth rate?  (b) What is the predicted population in 2015?  (c) What assumptions are being made in question (b)?  (d) When will the population reach 500 million? 11/23/2015 Precalculus - Lesson 21 - Exponential Models

20. 20 (E) Working with A = Pe rt Ex 3. A certain bacteria grows according to the formula A(t) = 5000e 0.4055t, where t is time in hours.  (a) What will the population be in 8 hours  (b) When will the population reach 1,000,000 Ex 4. The function P(t) = 1 - e -0.0479t gives the percentage of the population that has seen a new TV show t weeks after it goes on the air.  (a) What percentage of people have seen the show after 24 weeks?  (b) Approximately, when will 90% of the people have seen the show?  (c ) What happens to P(t) as t gets infinitely large? Why? Is this reasonable? 11/23/2015 Precalculus - Lesson 21 - Exponential Models

21. 21 (F) Logistic Models In real world applications, exponential models for growth have one limitation: think about the end behaviour of y = ca x (as x  ∞ ) populations cannot grow infinitely large => there are invariably factors that limit population growth (fixed area, food supply, etc…) So we have a function called a logistic model that accounts for the limitations imposed on a population: The equation is A is a constant reflecting the limit to growth, b is another constant, as is k If k > 0, then we have a GROWTH function If k < 0, then we have a DECAY function 11/23/2015 Precalculus - Lesson 21 - Exponential Models

22. 22 (F) Logistic Models Ex 1. The population of fish in a lake is given by the equation, where t is the number of years since 1975: (a) Find the initial population of fish (b) Find the population in 1987 (c ) Find when the population was 12,500 (d) Verify algebraically that there is an upper limit on the number of fish in the lake (limit as t  ∞ ) 11/23/2015 Precalculus - Lesson 21 - Exponential Models

23. 23 (F) Logistic Models Ex 2. Based on data from 1989 - 1994, the number of babies born each year in the US through assisted reproductive technology (ART) is approximated by the function, where t is the number of years since 1989 (a) In what year did the number of babies born through ART exceed 800? (b) In what year will the number of babies born through ART be 7500? (c) If this model remains accurate in the future, will the number of babies born through ART be 13,000? 11/23/2015 Precalculus - Lesson 21 - Exponential Models

24. (G) Modeling with Data Key Data feature  common RATIO between consecutive terms So that must mean our model must account for the RATIO between consecutive terms ….. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 24

25. (G) Modeling with Data (a) Graph the data on a scatter plot (b) How can you graphically analyze the data to help determine a model for the data? (c) How can you numerically analyze the data to help determine a model for the data? (d) Write an equation to model the data. Justify your choice of models. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 25

26. (H) Modeling Example #1 The following data table shows the relationship between the time (in hours after a rain storm in Manila) and the number of bacteria (#/mL of water) in water samples from the Pasig River: (a) Graph the data on a scatter plot (b) How can you graphically analyze the data to help determine a model for the data? (c) How can you numerically analyze the data to help determine a model for the data? (d) Write an equation to model the data. Justify your choice of models. Time (hours) # of Bacteria 0100 1196 2395 3806 41570 53154 66215 712600 825300 11/23/2015 Precalculus - Lesson 21 - Exponential Models 26

27. (H) Modeling Example #2 The value of my car is depreciating over time. I bought the car new in 2002 and the value of my car (in thousands) has been tabulated below: (a) Graph the data on a scatter plot (b) How can you graphically analyze the data to help determine a model for the data? (c) How can you numerically analyze the data to help determine a model for the data? (d) Write an equation to model the data. Justify your choice of models. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 27 Year200220032004200520062007200820092010 Value403632.429.226.223.621.319.117.2

28. (H) Modeling Example #3 The following data table shows the historic world population since 1950: (a) Graph the data on a scatter plot (b) How can you graphically analyze the data to help determine a model for the data? (c) How can you numerically analyze the data to help determine a model for the data? (d) Write an equation to model the data. Justify your choice of models. 11/23/2015 Precalculus - Lesson 21 - Exponential Models 28 Year195019601970198019901995200020052010 Pop (in millions) 2.563.043.714.455.295.7806.096.476.85