To find the GDP per capita of a country after 20 years given an initial GDP per capita and a steady growth rate, you need to follow these steps:
1. Identify the initial GDP per capita:
The initial GDP per capita is [tex]$66,000.
2. Determine the annual growth rate:
The annual growth rate is 3%. In decimal form, this is expressed as 0.03.
3. Identify the number of years over which the growth occurs:
The number of years is 20.
4. Use the formula for compound interest to calculate the final GDP per capita:
The formula to calculate GDP after a certain number of years with a steady growth rate is:
\[
\text{GDP}_{\text{final}} = \text{GDP}_{\text{initial}} \times (1 + r)^n
\]
where:
- \(\text{GDP}_{\text{final}}\) is the final GDP per capita.
- \(\text{GDP}_{\text{initial}}\) is the initial GDP per capita.
- \(r\) is the annual growth rate.
- \(n\) is the number of years.
5. Substitute the known values into the formula:
\[
\text{GDP}_{\text{final}} = 66{,}000 \times (1 + 0.03)^{20}
\]
6. Calculate the growth factor:
\[
1 + 0.03 = 1.03
\]
7. Raise the growth factor to the power of the number of years:
\[
1.03^{20}
\]
8. Multiply the initial GDP per capita by this result:
\[
66{,}000 \times 1.03^{20}
\]
9. Compute the result:
After performing the calculations, the GDP per capita after 20 years will be approximately:
\[
119{,}203.34
\]
Therefore, after 20 years of growing at a steady rate of 3%, the GDP per capita in the country would reach approximately $[/tex]119,203.34.
