# 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.

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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. 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 100r. y = C (1 + r) t

Finding the Balance in an Account 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

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 100r. 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 S OLUTION M ETHOD 2 U SE A F ORMULA Finding the Balance in an Account 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.

Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

So, the growth rate r is 2 and the percent of increase each year is 200%. 1 + r = 3 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? S OLUTION The population triples each year, so the growth factor is 3. 1 + r = 3 The population triples each year, so the growth factor is 3. Reminder: percent increase is 100r.

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? Writing an Exponential Growth Model 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.

A Model with a Large Growth Factor 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. t P 486060180540162020 512340 0 1000 2000 3000 4000 5000 6000 1723456 Time (years) Population P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase

W RITING E XPONENTIAL D ECAY M ODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. 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 100r. y = C (1 – r) t

Writing an Exponential Decay Model 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

Graphing the Decay of Purchasing Power 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. 0 0.2 0.4 0.6 0.8 1.0 112357911 Years From Now Purchasing Power (dollars) 246810 t y 0.8370.9650.9310.8990.8671.00 512340 0.70.8080.7790.7520.726 106789 Your dollar of today will be worth about 70 cents in ten years. Your dollar of today will be worth about 70 cents in ten years. y = 0.965 t Help

G RAPHING E XPONENTIAL D ECAY M ODELS E XPONENTIAL G ROWTH AND 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 C ONCEPT S UMMARY 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. (1 – r) is the decay factor, r is the decay rate. (1 + r) is the growth factor, r is the growth rate. (0, C)

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 100r. y = C (1 + r) t Back

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 100r. y = C (1 – r) t

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