Question 1171138:
In 2012, the population of a city was 5.21 million. The exponential growth rate was 2.86% per year.
a) Find the exponential growth function.
b) Estimate the population of the city in 2018.
c) When will the population of the city be 9 million?
d) Find the doubling time.
Found 2 solutions by Theo, ankor@dixie-net.com:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! general form of exponential formula is y = a * b^x
a is a constant.
b is the base
x is the exponent.
in this problem, a = 5.21 million and b = 1 + 2.86%/100 = 1.0286.
in 2012, the population was 5.21 million.
in 2018, the population will be 5.21 * 1.0286 ^ (2018 - 2012) = 5.21 * 1.0286 ^ 6 = 6.170450081 million.
to find when the population will be 9 million, your equation becomes:
9 = 5.21 * 1.0286 ^ x
divide both sides of this formula by 5.21 to get:
9/5.21 = 1.0286 ^ x
take the log of both sides of thie equation to get:
log(9/5.21) = log(1.0286^x)
since log(1.0286^x) = x * log(1.0286), the equation becomes:
log(9/5.21) = x * log(1.0286)
divide both sides of this equation by log(1.0286) to get:
log(9/5.21) / log(1.0286) = x
solve for x to get:
x = 19.38548963.
confirm by replace x in the original equation to get:
9 = 5.21 * 1.0286 ^ 19.38548963 becomes 9 = 9.
this confirms the value of x is good.
the population will grow to 9 million in 19.38548963 years.
to find when the population will double, the equation becomes:
2 = 1 * 1.0286 ^ x
simplify to get:
2 = 1.0286 ^ x
take the log of both sides of this equation to get:
log(2) = log(1.0286 ^ x)
since log(1.0286 ^ x) = x * log(1.0286), this equation becomes:
log(2) = x * log(1.0286)
solve for x to get:
x = log(2) / log(1.0286) = 24.5808602.
to confirm this is true, replace x in the original equation to get:
2 = 1.0286 ^ 24.5808602 which becomes 2 = 2.
this confirms the value of x is true.
the population will double in 24.5808602 years.
Answer by ankor@dixie-net.com(22740)  (Show Source):
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