SOLUTION: According to the U.S. Census Bureau, the population of the United States in 2008 was 304 million people. In addition, the population of the United States was growing at a rate of 1

Question 1201103: According to the U.S. Census Bureau, the population of the United States in 2008 was 304 million people. In addition, the population of the United States was growing at a rate of 1.1% per year. Assuming this growth rate is continues, the model
P(t)=304*(1.011)^t-2008
represents the population P (in millions of people) in year t.
According to the model, when will the population be 412 million people? Be sure to round your answer to the nearest whole year.
Year _____

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
population in 2008 = 304 million.
formula is f = p * (1+r) ^ n
when r = .011 (1.1%), the formula bcomes:
f = 304 * 1.011 ^ n
n is the number of years from 2008.
when f = 412 million, the formula becomes:
412 = 304 * 1.011 ^ n
divide both sides of the equation by 304 to get:
412/304 = 1.011 ^ n
take the log of both sides of this equation to get:
log(412/304) = log(1.011 ^ n)
by log rule that says log(b^n) = n * log(b), this becomes:
log(412/304) = n * log(1.011)
solve for n to get:
n = log(412/304) / log(1.011) = 27.78768868.
replace n in the original equation with that and solve for f to get:
f = 304 * 1.011 ^ 27.78768868 = 412.
this confirms the value of n is correct.
the population will grow to 412 million in 27.78768868 years after the end of 2008.
that would make it at the end of 2008 + that = at the end of 2035.787689.
that would be in the year 2036.
at the end of 2035, the population will be 304 * 1.011 ^ (2035 - 2008) = 408.4649392.
at the end of 2036, the population will be 304 * 1.011 ^ (2036 - 2008) = 412.9580535.
it becomes 412 some time between the end of 2035 and the end of 2036.

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