Answer :
To solve the problem of determining when Mexico's GDP will grow from [tex]$1 trillion to $[/tex]2 trillion given a 10% annual growth rate starting from 2016, let's follow these steps:
1. Understand the Growth Formula:
The economic growth can be modeled using the formula for compound growth:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + \text{growth rate})^n \][/tex]
where:
- Present Value (PV) = [tex]$1 \text{ trillion}$[/tex] (initial GDP in 2016)
- Future Value (FV) = [tex]$2 \text{ trillion}$[/tex] (target GDP)
- Growth rate = [tex]$10\% = 0.10$[/tex]
- \( n \) = number of years
2. Set Up the Equation:
We need to find \( n \) when the GDP reaches $2 trillion:
[tex]\[ 2 = 1 \times (1 + 0.10)^n \][/tex]
Simplifying this equation:
[tex]\[ 2 = (1.10)^n \][/tex]
3. Solve for \( n \):
To find \( n \), we can use logarithms:
[tex]\[ \log(2) = \log((1.10)^n) \][/tex]
Applying the power rule of logarithms:
[tex]\[ \log(2) = n \cdot \log(1.10) \][/tex]
Solving for \( n \):
[tex]\[ n = \frac{\log(2)}{\log(1.10)} \][/tex]
4. Compute \( n \):
Using common logarithmic values:
[tex]\[ \log(2) \approx 0.3010 \][/tex]
[tex]\[ \log(1.10) \approx 0.0414 \][/tex]
So:
[tex]\[ n = \frac{0.3010}{0.0414} \approx 7.27 \][/tex]
5. Determine the Year:
Since we are starting in 2016 and \( n \approx 7.27 \), the GDP will reach $2 trillion approximately 7 years after 2016:
[tex]\[ 2016 + 7 = 2023 \text{ (part of the 7th year)} \][/tex]
However, because 7.27 indicates a bit more than 7 years, it will actually be in the 8th year after 2016. Therefore, the year when the GDP reaches $2 trillion is:
[tex]\[ 2016 + 8 = 2024 \][/tex]
Thus, the correct answer is 2024, which is not listed in the provided options. Since the closest to this year in the multiple-choice options is between 2023 and 2027 but not precise, the accurate choice must be evaluated against other options if no other options closer to 2024 are given.
In this problem, since 2024 is our exact calculation, it's clear that the GDP will reach [tex]$\$[/tex] 2$ trillion by the year 2024.
1. Understand the Growth Formula:
The economic growth can be modeled using the formula for compound growth:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + \text{growth rate})^n \][/tex]
where:
- Present Value (PV) = [tex]$1 \text{ trillion}$[/tex] (initial GDP in 2016)
- Future Value (FV) = [tex]$2 \text{ trillion}$[/tex] (target GDP)
- Growth rate = [tex]$10\% = 0.10$[/tex]
- \( n \) = number of years
2. Set Up the Equation:
We need to find \( n \) when the GDP reaches $2 trillion:
[tex]\[ 2 = 1 \times (1 + 0.10)^n \][/tex]
Simplifying this equation:
[tex]\[ 2 = (1.10)^n \][/tex]
3. Solve for \( n \):
To find \( n \), we can use logarithms:
[tex]\[ \log(2) = \log((1.10)^n) \][/tex]
Applying the power rule of logarithms:
[tex]\[ \log(2) = n \cdot \log(1.10) \][/tex]
Solving for \( n \):
[tex]\[ n = \frac{\log(2)}{\log(1.10)} \][/tex]
4. Compute \( n \):
Using common logarithmic values:
[tex]\[ \log(2) \approx 0.3010 \][/tex]
[tex]\[ \log(1.10) \approx 0.0414 \][/tex]
So:
[tex]\[ n = \frac{0.3010}{0.0414} \approx 7.27 \][/tex]
5. Determine the Year:
Since we are starting in 2016 and \( n \approx 7.27 \), the GDP will reach $2 trillion approximately 7 years after 2016:
[tex]\[ 2016 + 7 = 2023 \text{ (part of the 7th year)} \][/tex]
However, because 7.27 indicates a bit more than 7 years, it will actually be in the 8th year after 2016. Therefore, the year when the GDP reaches $2 trillion is:
[tex]\[ 2016 + 8 = 2024 \][/tex]
Thus, the correct answer is 2024, which is not listed in the provided options. Since the closest to this year in the multiple-choice options is between 2023 and 2027 but not precise, the accurate choice must be evaluated against other options if no other options closer to 2024 are given.
In this problem, since 2024 is our exact calculation, it's clear that the GDP will reach [tex]$\$[/tex] 2$ trillion by the year 2024.
