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Sagot :
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.
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.
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