June 2022 Algebra 2 Regents, Part IV

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Algebra 2 June 2022

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.


37. The population, in millions of people, of the United States can be represented by the recursive formula below, where a0 represents the population in 1910 and n represents the number of years since 1910.

a₀ = 92.2
a_n = l.015a_n-1

Identify the percentage of the annual rate of growth from the equation a_n = l.015a_n-1.

Write an exponential function, P, where P(t) represents the United States population in millions of people, and tis the number of years since 1910.

According to this model, determine algebraically the number of years it takes for the population of the United States to be approximately 300 million people. Round your answer to the nearest year.

Answer:

The growth rate is 0.015, which is 1.5%.

The exponential function would be P(t) = 92.2(1.015)^t

To find the amount time until the population is approximately 300 million, set the function equal to 300 and solve for t.

300 = 92.2(1.015)^t

300/92.2 = (1.015)^t

300/92.2 = (1.015)^t

log(300/92.2) = t log(1.015)

log(300/92.2)/log(1.015) = t

t = 79.243...

Approximately 79 years.

Alternatively, you could have answered the last two parts as follows:

P(t) = 92.2e^.015t

300 = 92.2e^.015t

300/92.2 = e^.015t

ln(300/92.2) = .015t ln e

ln(300/92.2)/0.015 = t (1)

t = 78.6548...

Approximately 79 years.

Note that the decimal portions vary a little, but they both round to the same number of years.

End of Exam

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