SOLUTION: The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country during a year. During the period 1985-1999, the Gross Dom

Question 475306: The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country during a year. During the period 1985-1999, the Gross Domestic Product of the United States grew about 3.2% per year, measured in 1996 dollars. In 1985, the GDP was $577 billion.
In what year will the GDP reach $2 trillion? You just need to give the year, not part of a year.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
1985 to 1999 the GDP grew about 3.2% per year, measured in 1996 dollars.
In 1985, the GDP was $577 billion?
In what year will the GDP reach $2 trillion.
If the average growth rate is assumed to be the same as the average growth rate between 1985 and 1999, then the equation of when the value of the GDP will reach 1 trillion dollars is given by the equation:
f = p * (1.032)^n
f is the future value which is equal to 1 trillion dollars.
Since 1000 * 1 billion equals 1 trillion, we will make f = 1000 billion.
p is the present value which is equal to 577 billion.
n is the nuumber of years.
1.032 is the average increase in the value of the GDP per year.
3.2% is equal to a rate of .032 per year increase.
If the value in the present year is 1, then the value in the next year is 1 * 1.032 = 1.032^1 and the value in the year after that is 1 * 1.032 * 1.032 = 1 * 1.032^2.
We know that p = 577 billion.
We want f to equal 1000 billion.
We are solving for n.
The equation becomes:
1000 = 577 * (1.032)^n
divide both sides of this equation by 577 to get:
1000/577 = 1.032^n
take the log of both sides of this equation to get:
log(1000/577) = log(1.032^n)
since log(a^b) = b*log(a), this equation becomes:
log(1000/577) = n*log(1.032)
divide both sides of this equation by log(1.032) to get:
n = log(1000/577)/log(1.032)
use your calculator to get:
n = 17.45829471
The 577 billion will increase to 1000 billion in 17.45829471 years.
substitute in your original equation to confirm this is true.
the original equation is:
1000 = 577 * (1.032)^n which becomes:
1000 = 577 * (1.032)17.45829471 which becomes:
1000 = 1000, confirming the value for n of 17.45829471 is good.
The base year for the calculation is 1985.
add 17 to that and the future year for the calculation becomes 2002.
The GDP should reach 1 trillion in 2002 at the rate of 3.2% increase in GDP per year.
The general equation for this is:
y = 577 * 1.032^x
The graph of this equation looks like this:

The 2 horizontal lines are at 577 billion and 1000 billion.
577 billion is what is the value of y in 1985 (year 0).
1000 billion is what is the value of y somewhere between 2002 and 2003 (year 17 and year 18).
draw a vertical line from the intersection of the graph of the equation with the horizontal lines and you will see that the values of x are at 0 when y = 577 billion and are between 17 and 18 when y = 1000 billion, which is equivalent to 1 trillion.

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