1. C is the initial amount. t is the time period. (1 + r ) is the growth factor, r is the growth rate. The percent of increase is 100 r . y = C (1 + r ) t E XPONENTIAL G ROWTH M ODEL W RITING E XPONENTIAL G ROWTH M ODELS A quantity is growing exponentially if it increases by the same percent in each time period.
2. C OMPOUND I NTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? S OLUTION M ETHOD 1 S OLVE A S IMPLER P ROBLEM Find the account balance A 1 after 1 year and multiply by the growth factor to find the balance for each of the following years. The growth rate is 0.08 , so the growth factor is 1 + 0.08 = 1.08 . A 1 = 500 ( 1.08 ) = 540 Balance after one year A 2 = 500 ( 1.08 )( 1.08 ) = 583.20 Balance after two years A 3 = 500 ( 1.08 )( 1.08 )( 1.08 ) = 629.856 A 6 = 500 ( 1.08 ) 6 793.437 Balance after three years Balance after six years Finding the Balance in an Account • • • • • •
3. S OLUTION M ETHOD 2 U SE A F ORMULA C OMPOUND I NTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? Use the exponential growth model to find the account balance A . The growth rate is 0.08 . The initial value is 500 . E XPONENTIAL G ROWTH M ODEL C is the initial amount. t is the time period. (1 + r ) is the growth factor, r is the growth rate. The percent of increase is 100 r . y = C (1 + r ) t E XPONENTIAL G ROWTH M ODEL 500 is the initial amount. 6 is the time period. (1 + 0.08 ) is the growth factor, 0.08 is the growth rate. A 6 = 500 ( 1.08 ) 6 793.437 Balance after 6 years A 6 = 500 (1 + 0.08 ) 6 Finding the Balance in an Account
4. A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. Writing an Exponential Growth Model
5. So, the growth rate r is 2 and the percent of increase each year is 200%. 1 + r = 3 A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. a. What is the percent of increase each year? S OLUTION The population triples each year, so the growth factor is 3. 1 + r = 3 Reminder: percent increase is 100 r . So, the growth rate r is 2 and the percent of increase each year is 200%. So, the growth rate r is 2 and the percent of increase each year is 200% . 1 + r = 3 Writing an Exponential Growth Model The population triples each year, so the growth factor is 3.
6. A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. b. What is the population after 5 years? S OLUTION After 5 years, the population is P = C (1 + r ) t Exponential growth model = 20 (1 + 2 ) 5 = 20 • 3 5 = 4860 Help Substitute C , r , and t . Simplify. Evaluate. There will be about 4860 rabbits after 5 years. Writing an Exponential Growth Model
7. G RAPHING E XPONENTIAL G ROWTH M ODELS Graph the growth of the rabbit population. S OLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase A Model with a Large Growth Factor t P 4860 60 180 540 1620 20 5 1 2 3 4 0 0 1000 2000 3000 4000 5000 6000 1 7 2 3 4 5 6 Time (years) Population
8. A quantity is decreasing exponentially if it decreases by the same percent in each time period. C is the initial amount. t is the time period. (1 – r ) is the decay factor, r is the decay rate. The percent of decrease is 100 r . y = C (1 – r ) t W RITING E XPONENTIAL D ECAY M ODELS E XPONENTIAL D ECAY M ODEL
9. C OMPOUND I NTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? S OLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. = ( 1 )(1 – 0.035 ) t = 0.965 t y = C (1 – r ) t y = 0.965 15 Exponential decay model Substitute 1 for C , 0.035 for r . Simplify. Because 1997 is 15 years after 1982, substitute 15 for t . Substitute 15 for t . The purchasing power of a dollar in 1997 compared to 1982 was $0.59. 0.59 Writing an Exponential Decay Model
10. G RAPHING E XPONENTIAL D ECAY M ODELS Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. S OLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. Your dollar of today will be worth about 70 cents in ten years. y = 0.965 t Help Graphing the Decay of Purchasing Power 0 0.2 0.4 0.6 0.8 1.0 1 12 3 5 7 9 11 Years From Now Purchasing Power (dollars) 2 4 6 8 10 t y 0.837 0.965 0.931 0.899 0.867 1.00 5 1 2 3 4 0 0.7 0.808 0.779 0.752 0.726 10 6 7 8 9
11. G RAPHING E XPONENTIAL D ECAY M ODELS y = C (1 – r ) t y = C (1 + r ) t E XPONENTIAL G ROWTH M ODEL E XPONENTIAL D ECAY M ODEL 1 + r > 1 0 < 1 – r < 1 An exponential model y = a • b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1 . C is the initial amount. t is the time period. E XPONENTIAL G ROWTH AND D ECAY M ODELS C ONCEPT S UMMARY (1 – r ) is the decay factor, r is the decay rate. (1 + r ) is the growth factor, r is the growth rate. (0, C) (0, C)
12. Exponential Growth & Decay Models <ul><li>A 0 is the amount you start with, t is the time, and k=constant of growth (or decay) </li></ul><ul><li>f k>0, the amount is GROWING (getting larger), as in the money in a savings account that is having interest compounded over time </li></ul><ul><li>If k<0, the amount is SHRINKING (getting smaller), as in the amount of radioactive substance remaining after the substance decays over time </li></ul>
13. Graphs A 0 A 0
14. Example <ul><li>Population Growth of the United States . In 1990 the population in the United States was about 249 million and the exponential growth rate was 8% per decade. ( Source: U.S. Census Bureau ) </li></ul><ul><ul><li>Find the exponential growth function. </li></ul></ul><ul><ul><li>What will the population be in 2020? </li></ul></ul><ul><ul><li>After how long will the population be double what it was in 1990? </li></ul></ul>
15. Solution <ul><li>At t = 0 (1990), the population was about 249 million. We substitute 249 for A 0 and 0.08 for k to obtain the exponential growth function. </li></ul><ul><li>A( t ) = 249 e 0.08 t </li></ul><ul><li>In 2020, 3 decades later, t = 3. To find the population in 2020 we substitute 3 for t : </li></ul><ul><li>A(3) = 249 e 0.08(3) = 249 e 0.24 317. </li></ul><ul><li>The population will be approximately 317 million in 2020. </li></ul>
16. Solution continued <ul><li>We are looking for the doubling time T. </li></ul><ul><li>498 = 249 e 0.08 T </li></ul><ul><li>2 = e 0.08 T </li></ul><ul><li>ln 2 = ln e 0.08 T </li></ul><ul><li>ln 2 = 0.08 T (ln e x = x ) </li></ul><ul><li>= T </li></ul><ul><li>8.7 T </li></ul><ul><li>The population of the U.S. will double in about 8.7 decades or 87 years. This will be approximately in 2077. </li></ul>
17. Exponential Decay <ul><li>Decay, or decline, is represented by the function A( t ) = A 0 e kt , k < 0. </li></ul><ul><li>In this function: </li></ul><ul><li>A 0 = initial amount of the substance, </li></ul><ul><li>A = amount of the substance left after time, </li></ul><ul><li>t = time, </li></ul><ul><li>k = decay rate. </li></ul><ul><li>The half-life is the amount of time it takes for half of an amount of substance to decay. </li></ul>
18. Example <ul><li>Carbon Dating . The radioactive element carbon-14 has a half-life of 5715 years. If a piece of charcoal that had lost 7.3% of its original amount of carbon, was discovered from an ancient campsite, how could the age of the charcoal be determined? </li></ul><ul><li>Solution : The function for carbon dating is </li></ul><ul><li>A( t ) = A 0 e -0.00012t . </li></ul><ul><li>If the charcoal has lost 7.3% of its carbon-14 from its initial amount A 0 , then 92.7%A 0 is the amount present. </li></ul>
19. Example continued <ul><li>To find the age of the charcoal, we solve the equation for t : </li></ul><ul><li>The charcoal was about 632 years old. </li></ul>
20. E XPONENTIAL G ROWTH M ODEL C is the initial amount. t is the time period. (1 + r ) is the growth factor, r is the growth rate. The percent of increase is 100 r . y = C (1 + r ) t
21. E XPONENTIAL D ECAY M ODEL C is the initial amount. t is the time period. (1 – r ) is the decay factor, r is the decay rate. The percent of decrease is 100 r . y = C (1 – r ) t
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