Question 703248
Two neighboring towns have had population changes over a ten year period that follow exponential growth or exponential decay patterns.- The population of Town A was 50,000 in 1980. It has increased in population by approximately 5.1% per year.- The population of Town B was 100,000 in 1980. It has experienced a decrease in population of 8.1% per year. 
Part A: Write an exponential model to describe the population of Town A. Estimate the population in the year 1988.
P(x) = 50,000*(1+0.051)^x where x is number of years after 1980
P(8) = 50,000*(1.051)^8 is approximately 74,438
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Part B: Write an exponential model to describe the population of Town B.
P(x) = 100,000*(1-0.081)^x
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Part C: Estimate the population of town B in the year 2005. Is this a good approximation? Explain why or why not.
P(5) = 100,000*(0.919)^5 = 65,551
Looks good to me.
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Part D: Which model represents exponential growth?:::: A
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Which is a model of exponential decay?::: B
 Explain why exponential growth or exponential decay models can be used for this data.
A is increasing exponentially ; B is decreasing exponentially.
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Part E: Approximately how many years would it take the population of Town A to double? Determine the solution algebraically.
Solve: (1.051)^x = 2
x = log(2)/log(1.051) is approximately 14 years
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Part F: In how many years would the population of Town B decrease by 25%? Determine the solution algebraically.
Solve (0.919)^x = 0.75
x = log(0.75)/log(0.919) is more than 3 years and less than 4
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Part G: Determine the year when the towns would have approximately the same population. Use two different methods. Explain why using an algebraic method would be difficult here.
50,000(1.051)^x = 100,000(0.919)^x
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(1.051/0.919)^x = 2
x = log(2)/log(1.144) is approximately 5.165 or more than 5 years; less than 6
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Cheers,
Stan H.