Exponential Growth and Decay

1. Exponential Growth and Decay Section 6.1

2. Given x = 2, y = ½, and z = 4.1 • Evaluate each expression: • 1.) 3y • 2.) 42z • 3.) 10(2)y+2 • 4.) -5yz

3. You can use a calculator to model the growth of 25 bacteria, assuming that the entire population doubles every hour.

4. Some things grow or decay at an exponential rate rather than a steady (linear) rate. • This means they grow or decay very rapidly. • To find the amount after a certain time period you must know 3 things: • 1. original amount • 2. growth/decay rate • 3. Time period of growth/decay

5. Use this basic expression to write an expression for exponential growth/decay. • (Original population)(growth)number of time periods • Ex. The 25 bacteria in the original problem doubles every hour, find the number of bacteria after 10 hours • Original population = 25 • Growth = 2 (since it doubles) • Time periods = 10 (since it doubles each hour) • Expression: • Population after 10 hours = 25, 600

6. Ex. 2 If 50 bacteria triple every 3 hours, find the number of bacteria after 12 hours. • Original amount = 50 • Growth = 3 • Time periods = 4 (every 3 hours it triples 12/3 = 4) • Expression: • Answer: 4050

7. Ex. 3 200 bacteria double every 15 minutes, find the amount after 1.5 hours. • Original amount: 200 • Growth: 2 • Time period: 6 (90 minutes/ 15 minutes) • Expression: • Answer: 12, 800

8. Ex. 4 100 bacteria triple every 2 hours. Find the amount after 5 hours. • Original = 100 • Growth = 3 • Time periods = 5/2 • Expression : • Answer 1558.845… rounds to 1559

9. From yesterday, • Use this basic expression to write an expression for exponential growth/decay. • (Original population)(growth)number of time periods • Assuming an initial population of 100 bacteria, predict the population of bacteria after n hours if the population doubles. • The population after n hours can be represented by the following exponential expression: • is called an exponential expression because the exponent, n, is a variable and the base, 2, is a fixed number. • The base of an exponential expression is called the multiplier.

10. To find a multiplier • Add or subtract the growth decay rate from 100% • Change to a decimal. • Ex 1. • 5.5% growth • 100% + 5.5% = 105.5% • Multiplier = 1.055

11. Ex. 2 • 0.25 growth • 100% + 0.25% = 100.25% • Multiplier = 1.0025 • Ex. 3 • 3% decay • 100% - 3% = 97% • Multiplier = .97

12. Ex. 4 • 0.5% decay • 100% - 0.5% = 99.5% • Multiplier = .995

13. Modeling Growth or Decay • Ex. 1 Since 1990, the population of the United States has been growing at a rate of 8% each decade. If the population was 248,718,301 in 1990, predict the number of people in 2020. Round to the nearest hundred thousand. • Original amount = 248,718,301 • Multiplier = 1.08 (100% + 8%) • Time period = 3 (30 years/10 years) • Expression: • Answer: 313,313,428.4 • 313, 300, 000 people

14. Suppose you buy a car for $15,000. Its value decreases at a rate of about 8% per year. Predict the value of the car after 4 years, and after 7 years. • Solution: • Multiplier: 100% - 8% = 92% = .92 • Exponential expression: • Value after 4 years: • Value after 7 years: $8367.70

15. Ex. 2 You invested $1000 in a company’s stock at the end of 2009 and that stock has increased at a rate of about 15% per year. Predict the value of the stock, to the nearest cent, at the end of the years 2014. • Original amount: 1000 • Multiplier: 1.15 (100% + 15%) • Time periods: 5 (5 years / 1 year) • Expression: • Answer: $2011.36

16. Ex. 3 You buy a new car for $15,000 and its value decreases at a rate of about 8% per year. Predict the value of the car, to the nearest cent, after 4 years. • Original amount: 15,000 • Multiplier: .92 (100% - 8%) • Time: 4 • Expression: • Answer: $10, 745.89