Question 1161308: Suppose that the population in a small city is 39,000 in the beginning of 2012 and the city council assumes that the population size t years later can be estimated by the equation
P=39,000e^0.09t
Approximately when will the city have a population of 60,000?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the equation is p = 39,000 * e^(.09*t)
when the population is 60,000, the equation becomes:
60,000 = 39,000 * e^(.09*t)
divide both sides of the equation by 39,000 and simplify to get:
60/39 = e^(.09*t)
take the natural log of both sides of the equation to get:
ln(60/39) = ln(e^(.09*t))
by the properties of logarithms, this is equivalent to:
ln(60/39) = .09*t*ln(e)
since ln(e) = 1, this becomes:
ln(60/39) = .09*t
divide both sides of this equation by .09 and solve for t to get:
t = ln(60/39)/.09 = 4.786476845
confirm by replacing t in the original equation to get:
f = 39,000 * e^(.09*4.786476845) = 60,000
this confirms the solution is correct.
since 39,000 population is in the beginning of 2012, and the population reaches 60,000 4.786476845 years from then, the population reaches 60,000 sometime in the year 2016.
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