Exponential Modeling Growth and Decay
Identify each as growth or Decay What you should Know
y Exponential functions 0<b<1 decay b>1 Growth Review
WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. EXPONENTIAL GROWTH MODEL a is the initial amount. x is the time period. y = a (1 + r) x (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r.
Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 1 SOLVE A SIMPLER PROBLEM 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 A 2 = 500(1.08)(1.08) = 583.20 A 3 = 500(1.08)(1.08)(1.08) = 629.856 A 6 = 500(1.08) 6 793.437 Balance after one year Balance after two years Balance after three years Balance after six years
Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 2 USE A FORMULA Use the exponential growth model to find the account balance A. The growth rate is 0.08. The initial value is 500. EXPONENTIAL GROWTH MODEL C 500 is the is the initial initial amount. 6 t is is the time period. y = C (1 + r) t A 6 = 500 (1 + 0.08) (1 + (1 0.08) + r) is is the the growth factor, factor, 0.08 r is the growth rate. rate. The percent of increase is 100r. A 6 = 500(1.08) 6 793.437 Balance after 6 years 6
Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
Writing an Exponential Growth Model 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? SOLUTION The population triples each year, so the growth factor is 3. 1 + r = 3 So, the growth rate r is 2 and the percent of increase each year is 200%. Reminder: percent increase is 100r.
Writing an Exponential Growth Model 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? Help SOLUTION After 5 years, the population is P = C(1 + r) t Exponential growth model = 20(1 + 2) 5 = 20 35 Substitute C, r, and t. Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years.
Population GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor Graph the growth of the rabbit population. SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t 0 1 2 3 4 5 P 20 60 180 540 1620 4860 6000 5000 4000 3000 2000 1000 P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase 0 1 2 3 4 5 6 7 Time (years)
WRITING EXPONENTIAL DECAY MODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. a is the initial amount. EXPONENTIAL DECAY MODEL x is the time period. y = a (1 r) x (1 r ) is the decay factor, r is the decay rate. The percent of decrease is 100r.
Writing an Exponential Decay Model COMPOUND INTEREST 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? SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. y = C (1 r) t = (1)(1 0.035) t = 0.965 t Exponential decay model Substitute 1 for C, 0.035 for r. Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.965 15 Substitute 15 for t. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
Purchasing Power (dollars) GRAPHING EXPONENTIAL DECAY MODELS Graphing the Decay of Purchasing Power Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. Help SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t 0 1 2 3 4 y 1.00 0.965 0.931 0.899 0.867 0.837 5 6 7 8 9 0.808 0.779 0.752 0.726 10 0.7 1.0 0.8 0.6 0.4 0.2 0 Your dollar of today will be worth about 70 cents in ten years. y = 0.965 t 1 2 3 4 5 6 7 8 9 10 11 12 Years From Now
GRAPHING EXPONENTIAL DECAY MODELS CONCEPT SUMMARY EXPONENTIAL GROWTH AND DECAY MODELS EXPONENTIAL GROWTH MODEL y = C (1 + r) t EXPONENTIAL DECAY MODEL y = C (1 r) t An exponential model y = a b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. (1 + r) is the growth factor, (1 r) is the decay factor, r is the growth C t is the initial time period. amount. (0, C) rate. r is the decay (0, rate. C) 1 + r > 1 0 < 1 r < 1
Read and take notes on Methods and Meanings, on page 83 Do the problems: Review and Preview Page 84 #86-91 On your own