SOLUTION: Modeling Population: The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential function

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Question 703916: Modeling Population: The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula A = 16.6 e to the power of 0.0547t models the population of a US state, A, in millions, t years after 2000.
a. What was the population in 2000?
b. When will the population of the state reach 23.3 million?
Answer by josgarithmetic(39838) About Me  (Show Source):
You can put this solution on YOUR website!
Description of your model is A=16.6e%5E%280.0547%2At%29

a. At year 2000, t=0, since t was described as years after 2000.
A=16.6e%5E%280.0547%2A0%29
A=16.6e%5E0=16.6%2A1=16.6, million

b. When will A=23.3 million?
You want to use the formula, solved for t.
If you really need the steps, they may be supplied, but I will tentatively omit them here; **See below. you would begin by taking natural log of both sides of A=16.6e%5E%280.0547%2At%29.
Your formula for t will be or equivalent to:
t=%28ln%2816.6%29-ln%28A%29%29%2F%280.0547%29
Substitute your desired 23.3 for A and find t.

**From formula for A,
ln%28A%29=ln%2816.6%2Ae%5E%280.0547t%29%29
ln%28A%29=ln%2816.6%29%2Bln%28e%5E%280.0547t%29%29
ln%28A%29=ln%2816.6%29%2B0.0547%2At%2Aln%28e%29
ln%28A%29=ln%2816.6%29%2B0.0547%2At%7D%7D%0D%0A%7B%7B%7Bln%28A%29-ln%2816.6%29=0.0547%2At
t=%28ln%28A%29-ln%2816.6%29%29%2F0.0547