Algebra 2 Trig/Apps
Name:
Lesson #48
Objective
Solve problems involving exponential rgowth and decay
DO NOW
Simplify each expression.
1. (4 + 0.05)2
2. 25(1 + 0.02)3
3. 1 +
0.03
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4. The initial value of an exponential function is 3 and the common ratio is 2.
What is the 5th term?
5. The function f(x) = 2(4)x models an insect population after x days. What is the
population after 3 days?
Exponential growth occurs when a quantity increases by the same rate r in
each period t.
Exponential decay occurs when a quantity decreases by the same rate r in
each period t.
y = the final amount
a = initial amount
r = the growth or decay rate, expressed as a decimal
t = the amount of time that has passed
Then an exponential growth function is represented by y = a(1 + r)t
and an exponential decay function is represented by y = a(1 – r)t
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Example 1: Exponential Growth
The original value of a painting is $9,000 and the value increases by 7% each
year. Write an exponential growth function to model this situation. Then find
the painting’s value in 15 years.
Practice 1
A sculpture is increasing in value at a rate of 8% per year, and its value in 2000
was $1200. Write an exponential growth function to model this situation. Then
find the sculpture’s value in 2006.
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Example 2
The function f(x) = 500(1.035)x models the amount of money in a certificate of
deposit after x years.
a. What is the value of a? What does it indicate in terms of the problems?
b. What is the value of b? What does it indicate in terms of the problem?
c. Is the curve exponential GROWTH or DECAY? How can you tell?
d. How much money will there be in 6 years?
Practice 2
In 2000, each person in India consumed an average of 13 kg of sugar. Sugar
consumption in India is projected to increase by 3.6% per year.
a. Write an equation that represents the average sugar consumption for people
in India, where x represents the number of years past the year 2000.
b. Using this model, in about what year will sugar consumption average about
18 kg per person?
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Example 3: Exponential Decay
The function f(x) = 8(0.75)x models the width of a photograph in inches after it
has been reduced by 25% x times.
a. What is the value of a?
b. What is the value of b?
c. How can you tell that the function is exponential decay?
What is the rate of decay?
d. What is the width of the photograph after it has been reduced 3 times?
Practice 3
The fish population in a local stream is decrasing at a rate of 3% per year. The
original population was 48,000.
a. Write an exponential decay function to model this situation.
b. Find the population after 7 years.
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HW # 48
1. The function y = 11.6(1.009)x models residential energy consumption in
quadrillion Btu where x is the number of years after 2003.
a. Does this equation represent exponential growth or decay? How can you
tell?
b. What will residential energy consumption be in 2013?
2. In 2000, the population of Texas was about 21 million, and it was growing
by about 2% per year.
a. If y = the number of millions of people, and x is the number of years
after 2000, what is the equation that models this situation?
b. Using this model, in about what year will the population reach 30 million?
(use trial and error with various values of x.)
3. The population of a town is decreasing at a rate of 3% per year. In 2000
there were 1700 people. Let x = # of years after the year 2000.
a. Write an exponential decay function to model this situation.
b. Find the population in 2012.
4. A population of wolves in a county is represented by the equation
P(t) = 80(0.98)t, where t is the number of years since 1998. What was the wolf population
in the year 2008?
5. On January 1, 1999, the price of gasoline was $1.39 per gallon. If the price of gasoline
increased by 0.5% per month, what was the cost of one gallon of gasoline, to the nearest
cent, on January 1st 12 months later?
6. Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates (loses
value) at a rate of 5%. What will its approximate value be at the end of the fourth year?
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7. A used car was purchased in July 1999 for $11,900. If the car depreciates 13% of its value
each year, what is the value of the car, to the nearest hundred dollars, in July 2002?
8. The number of houses in Central Village, New York, was 540 in the year 1995. New
houses are being built at a rate of 3.9%.
a. Write an equation that represents the number of houses each year.
b. Determine the number of houses in Central Village in the year 2010.
9. Since January 1980, the population of the city of Brownville has grown according to
the mathematical model y = 720,500(1.022)x, where x is the number of years since
January 1980.
a. Explain what the number 720,500 and 1.022 represent in this model.
b. In what year will the population reach 1,548,800?
10. The amount of Tylenol left in the body decreases at a rate of 20% per hour. If the initial
dosage of Tylenol is 10 milligrams, how much Tylenol will be in the body after 8 hours?
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EXIT TICKET
Name:
Date:
The number of employees at a certain company is 1440 and is increasing at a rate of 1.5%
per year. Write an exponential growth function to model this situation. Then find the number
of employees in the company after 9 years.
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EXIT TICKET
Name:
Date:
The number of employees at a certain company is 1440 and is increasing at a rate of 1.5%
per year. Write an exponential growth function to model this situation. Then find the number
of employees in the company after 9 years.
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