Answer:
a. 18.5 million
b. 2002
Explanation:
Given that A = 18.5·e^(0.1708t) models a population t years after 2000, you want to know the population in 2000 and when it will reach 26.6 million.
Population in 2000
The year 2000 is 0 years after 2000, so we can find the population using t=0 in the given equation.
A = 18.5·e^(0.1708·0) = 18.5·e^0 = 18.5
The population in 2000 was 18.5 million.
Year of 26.6 million
We can solve for t to find the year in which the population reached 26.6 million:
26.6 = 18.5·e^(0.1708t)
26.6/18.5 = e^(0.1708t)
ln(266/185) = 0.1708t
t = ln(266/185)/0.1708 ≈ 2.13 . . . . . . 2.13 years after 2000
The population will reach 26.6 million in the year 2002.
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Additional comment
The population of 26.6 million represents about a 44% increase over the initial population of 18.5 million. The exponential term tells you the rate of growth is 17.08% compounded continuously. Thus we expect the larger population to be reached in a time slightly less than 44/17 ≈ 2.6 years.
(These numbers were found using a calculator. It is sufficient to do the estimating by realizing that 1.50·18 = 27, and 17·3 = 51, so the growth to 26.6 from 18.5 is less than 50% and will take less than 3 years.)
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