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Exponential Growth and Decay Equations in Various Scenarios

This lesson covers the general equations for exponential growth and decay, and applies them to different scenarios such as population growth, compound interest, coal usage, charity donations, and depreciation of assets.

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Exponential Growth and Decay Equations in Various Scenarios

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  1. Lesson 10-6 Growth and Decay

  2. Key Concept The general equation for exponential growth in which the initial amount C increases by the same percent over a given period of time is y = C(1 + r)t. In 1971, there were 294,105 females participating in high school sports. Since then, that number has increased an average of 8.5% per year. Write an equation to represent the number of females participating in high school sports since 1971. y = C (1 + r)t General equation for exponential growth y = 294,105 (1 + 0.085)t C = 294,105 and r = 8.5% or 0.085. y = 294,105(1.085)t. Simplify. y represents number of female athletes and t represents years since 1971.

  3. POPULATION: In 2000, the United States had a population of about 280 million, and a growth rate of about 0.85% per year. Write an equation to represent the population of the United States since the year 2000. y = 280,000,000(1.0085)t According to the equation, what will be the population of the United States in the year 2010? About 304,731,295

  4. Key Concept The equation for compound interest is . A represents the amount of investment, P is the principal(initial amount of investment),r represents the annual rate of interestexpressed as a decimal, n represents the number of times times the money is compounded each year, and t represents the number of years that the money is invested. • Native Americans received $24 for Manhattan in 1626. If the money had been invested at 6% per year compounded semiannually, how much money would there be in the year 2026? • Compound interest equation. • P = 24, r = 6% or 0.06, n = 2, and t = 400 • Simplify • There would be about $447,000,000,000

  5. Compound Interest: When Jing May was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. The money will go to Jing May when she turns 18 to help with her college expenses. What amount of money will Jing May receive from the investment? About $3380

  6. Key Concept The general equation for exponential decay where y represents the final amount,C represents the initial amount, r represents the rate of decay, and t represents time is y = C(1 - r)t. In 1950, the use of coal by residential and commercial users was 114.6 million tons. Many businesses now use cleaner sources of energy. As a result, the use of coal has decreased by 6.6% per year. Write an equation to represent the use of coal since 1950. y = C (1 - r)t General equation for exponential decay. y = 114.6 (1 - 0.066)t C = 114.6 and r = 6.6% or 0.066. y = 114.6(0.934)t. Simplify. y represents coal used annually and t represents years since 1950. Estimate the estimated amount of coal that will be used in 2015. Write an equation to represent the use of coal since 1950. y = 114.6 (0.934)t Equation for coal use. y = 114.6 (0.934)65 t = 2015-1950 = 65 y 1.35 The amount of coal should be about 1.35 million tons.

  7. CHARITY: During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were $390,000. Write an equation to represent the charity’s donations since the beginning of the recession. A = 390,000(0.989)t Estimate the amount of the donations 5 years after the start of the recession. About $369,017

  8. Ex. 4 Depreciation: FARMING: A farmer buys a tractor for $50,000. If the tractor depreciates 10% per year, find the value of the tractor in 7 years. y = C(1 - r)t General equation for exponential decay y = 50,000(1 - 0.10)7 C = 50,000, r = 10% or 0.10, and t = 7 y = 50,000(0.90)7Simplify y 23914.85 Use a calculator. The tractor will be worth about $23,914.85 or less than half its original value.

  9. Depreciation: Jackson and Elizabeth bought a house when they were first married ten years ago. Since that time the value of the real estate in their neighborhood has declined 3% per year. If they initially paid $179,000 for their house, what it its value today? About $131,999

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