8.1 Exponential Growth and Decay
8.2 Exponential Decay
• An exponential function involves the expression bx, where the base b is a positive number other than 1.
• If y = abx and a > 0 and b > 1, then the equation represents
_________________________________
• If y = abx and a > 0 and 0 < b < 1, then the equation represents __________________________________
• The general form of an exponential model is: y = a(1 ± r)2, where a is the _________________, r is the _______________________. The quantity (1 + r) is called the __________________________ while the quantity (1 ­ r) is the ___________________________.
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Examples ­ For Us
(1) In January 1993, there were about 1, 310, 000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. Write a model given the number h (in millions) of hosts t years after 1993. About how many hosts were there in 1996?
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(2) You buy a new car for $24, 000. The value y of the car decreases by 16% each year. Write a model for the value of the car after t years. Use the model to estimate the value after 2 years.
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Examples ­ For You
(3) There are 40, 000 homes in your city. Each year 10% of the homes are expected to disconnect from the septic systems and connect to the sewer system. Write a model for the number of homes that still use septic systems. Use the model to estimate the number of homes using septic systems after 5 years.
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(4) In 1980 about 2, 180, 000 US workers worked at home. During the next ten years, the number of workers working at home increased 5% per year. Write a model given the number w (in millions) of workers working at home t years after 1980.
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Guided Practice
(1) The population of Winnemucca, Nevada, can be modeled by P = 6191(1.04)t, where t is the number of years since 1990. What was the population in 1990? By what percent did the population increase each year?
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(2) The amount g (in trillions of cubic feet) of natural gas consumed in the US from 1940 to 1970 can be modeled by: g = 2.91(1.07)t, where t is the number of years since 1940.
a) Identify the initial amount , the growth factor, and the annual percent increase.
b) Estimate the natural gas consumption in 1955.
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(3) The number A (in millions) of record albums sold each year in the US from 1982 to 1993 can be modeled by A = 265(0.39)t, where t represents the number of years since 1982.
a) Identify the initial amount, the decay factor, and the annual percent decrease.
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In 4 ­ 6, write an exponential growth or decay model for the scenario described. Then answer the corresponding question.
(4) You buy a commemorative coin for $110. Each year t, the value V of the coin increases by 4%.
(5) You buy a stereo system for $780. Each year t, the value V of the stereo system decrease by 5%. Write an exponential decay model that describes the situation.
(6) You drink a beverage with 120 milligrams of caffeine. Each hour h, the amount of c of caffeine in your system decrease by about 12%. Write an exponential decay model that describes the situation.
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Compound Interest
Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal), compound n times per year. The amount A in the account after t years can be modeled by this equation:
A = P
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Examples ­ For Us
(1) You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency.
a) Annually
b) Quarterly
c) Daily
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Examples ­ For You
(1) You deposit $1600 in a bank account. Find the balance after 3 years for each of the following situations:
a) The account pays 2.5% annual interest compound annually.
b) The account pays 1.75% annual interest compound quarterly.
c) The account pays 4% annual interest compound yearly.
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Take out your student handbook and write down
the homework assignment:
WS: 8.1, 8.2
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