Sure! Let's go through the steps to determine when Mexico's GDP will double from [tex]$1 trillion to $[/tex]2 trillion with an annual growth rate of 10%:
1. Define the initial values:
- Initial GDP in 2016: \(1\) trillion dollars.
- Target GDP: \(2\) trillion dollars.
2. Understand the growth rate:
- Annual growth rate: \(10\%\) or \(0.10\).
3. Determine the formula to use for exponential growth:
The formula for future value with continuous growth is:
[tex]\[
GDP_{\text{future}} = GDP_{\text{initial}} \times (1 + \text{growth\_rate})^t
\][/tex]
where \(t\) is the number of years, \(GDP_{\text{initial}}\) is the initial GDP, and \(\text{growth\_rate}\) is the rate of growth.
4. Set up the equation with the values given:
[tex]\[
2 = 1 \times (1 + 0.10)^t
\][/tex]
Simplifying, we get:
[tex]\[
2 = (1.10)^t
\][/tex]
5. Solve for \(t\) using logarithms:
Taking the natural logarithm on both sides:
[tex]\[
\ln(2) = \ln((1.10)^t)
\][/tex]
Using the property of logarithms \(\ln(a^b) = b \cdot \ln(a)\), we get:
[tex]\[
\ln(2) = t \cdot \ln(1.10)
\][/tex]
Solving for \(t\):
[tex]\[
t = \frac{\ln(2)}{\ln(1.10)}
\][/tex]
6. Calculate the values:
- Using logarithm values: \(\ln(2) \approx 0.693\) and \(\ln(1.10) \approx 0.095\).
[tex]\[
t \approx \frac{0.693}{0.095} \approx 7.27
\][/tex]
So, it will take approximately \(7.27\) years for the GDP to double.
7. Determine the target year:
- Initial year: \(2016\)
- Time to double: \(7.27\) years
Adding the 7.27 years to 2016, we get approximately the year:
[tex]\[
2016 + 7.27 \approx 2023.27
\][/tex]
Since we need to consider whole years, we round up to the next full year, giving us \(2024\).
Therefore, the GDP of Mexico is expected to reach $2 trillion by the year 2024. The answer closest to this outcome is:
1. 2023
2. 2027
3. 2030
4. 2032
The correct answer is none of those options given perfectly align with 2024, but the calculated accurate year is 2024.
