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x function WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. y = C (1 + r)t (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation y C (1 r ) t Where C = initial amount r = growth rate (percent written as a decimal) t = time where t 0 (1+r) = growth factor where 1 + r > 1 You deposit $1500 in an account that pays 2.3% interest compounded yearly, 1) What was the initial principal (P) invested? 2) What is the growth rate (r)? The growth factor? 3) Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money? 1) The initial principal (P) is $1500. 2) The growth rate (r) is 0.023. The growth factor is 1.023. 3) A P (1 r ) t A 1500 (1 0.023) 2 A $1569.79 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. b. What is the population after 5 years? SOLUTION After 5 years, the population is P = C(1 + r) t Exponential growth model = 20(1 + 2) 5 Substitute C, r, and t. = 20 • 3 5 Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years. Help GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor Graph the growth of the rabbit population. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. SOLUTION t 0 P 20 1 2 3 60 180 540 4 5 1620 4860 6000 Population 5000 4000 P = 20 ( 3 ) t 3000 Here, the large growth factor of 3 corresponds to a rapid increase 2000 1000 0 1 2 3 4 Time (years) 5 6 7