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## EXPONENTIAL FUNCTIONS

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**1 +r > 1**A quantity is growing exponentially if it increases by the same percent in each unit of time. This is called exponential growth. C is the initial amount (1 + r) is the growth factor y = C(1 + r)t r is the growth rate t is the time period**You deposit $500 in an account that pays 8% annual interest**compounded yearly. What is the account balance after 6 years? y = C(1 + r)t y = 500(1 + 0.08)6 Note: always convert the percentage into a decimal. = 500(1.08)6 793.437 The balance after 6 years will be about $793.44.**Graphically**793.437 t The x-axis usually represents time**A savings account certificate of $1000 pays 6.5% annual**interest compounded yearly. What is the balance when the certificate matures after 5 years? y = C(1 + r)t y = 1000(1 + 0.065)5 = 1000(1.065)5 1370.09 The balance of the certificate after 5 years is about $1370.09.**A newly hatched channel catfish typically weighs about 0.3**grams. During the first six weeks of life, its growth is approximately exponential, increasing by about 10% each day. Find the weight at the end of six weeks. y = C(1 + r)t y = 0.3(1 + .10)42 Note: since the problem stated that the rate of increase was daily, we must let t be represented in days. y = 0.3(1.1)42 y 16.42910977 The weight is about 16.4 grams.**An experiment started with 100 bacteria. They double in**number every hour. Find the number of bacteria after 8 hours. y = C(1 + r)t y = 100(1 + 1)8 Note: since the bacteria doubled every hour, they rate of increase was 100% or 1. y = 100(2)8 y = 25,600 The number of bacteria after 8 hours is 25,600.**A population of 20 rabbits is released into a wild-life**region. The population triples each year for 5 years. • What is the percent increase each year? • What is the population after 5 years? Solution a. The population triples each year, so the growth factor is 3. 1 + r = 3 • So, the growth factor rate, r, is 2 and the percent increase each year is 200%.**A population of 20 rabbits is released into a wild-life**region. The population triples each year for 5 years. • What is the percent increase each year? • What is the population after 5 years? Solution There will be about 4860 rabbits after 5 years. • P = C(1 + r)t • = 20(1 + 2)5 • = 20(3)5 • = 4860**Graphically**4860 t**A company starts with 50 employees and after one year has**75. It increases the employees at the same rate every year for 4 years. • What is the percent of increase each year? • Find the number of employees after 4 years? solution • After one year you have 75 employees which means there • are 25 more employees than the company started with. • Since there were 50 employees and 25 were added within • one year, the percent increase must be 50%. Try this example on a sheet of paper before clicking.**A company starts with 50 employees and after one year has**75. It increases the employees at the same rate every year for 4 years. • What is the percent of increase each year? • Find the number of employees after 4 years? solution There will be 253 employees at the end of 4 years. • P = C(1 + r)t • = 50(1 + 0.5)4 • = 50(1.5)4 • 253**0 < 1 – r < 1**A quantity is decaying exponentially if it decreases by the same percent in each unit of time. This is called exponential decay. C is the initial amount (1 – r) is the decay factor y = C(1 r)t r is the decay rate t is the time period**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 (Notice the dollar decreased, therefore, we use a decay model). y = C(1 –r)t Because 1997 is 15 years after 1982, substitute 15 for t. y = 1(1 – 0.035)15 0.59**You bought a used car for $18,000. The value of the car**will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. • Write an exponential decay model to represent this situation. • Estimate the value of your car in 8 years. solution • The initial value C is $18,000. The decay rate r is 0.12. Let • tbe the age of the car in years. y = 18,000(1 – 0.12)t y = 18,000(0.88)t**You bought a used car for $18,000. The value of the car**will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. • Write an exponential decay model to represent this situation. • Estimate the value of your car in 8 years. solution b. To find the value in 8 years, substitute 8 for t. y = 18,000(0.88)8 6473 The value of your car in 8 years will be about $6473.**Checkpoint**A house costs $140,000 in the year 2002. It appreciates 4.5% annually. • Write a model to represent the value of the house • after t years. • b. Find the value of the house after 20 years. The amount of a particular medicine in the body decreases at a rate of 40% an hour. You are given 250mg. • Write a model to represent the amount of medicine • in the body after t hours. • b. Find the amount of medicine after 5 hours.