Question
The exponential model Upper A equals 826.9 e Superscript 0.017 t describes the population, A, of a country in millions, t years after 2003. Use the model to determine when the population of the country will be 916 million.
Question content area bottom
Part 1
The population of the country will be 916 million in
enter your response here.
(Round to the nearest year as needed.)
Question content area bottom
Part 1
The population of the country will be 916 million in
enter your response here.
(Round to the nearest year as needed.)
Answers
GPT-4o mini
To find out when the population of the country will reach 916 million using the given model \( A = 826.9 \cdot e^{0.017t} \), we need to set \( A \) to 916 and solve for \( t \).
1. Set up the equation:
\[
916 = 826.9 \cdot e^{0.017t}
\]
2. Divide both sides by 826.9 to isolate the exponential term:
\[
\frac{916}{826.9} = e^{0.017t}
\]
Calculating the left side:
\[
\frac{916}{826.9} \approx 1.107
\]
3. Now take the natural logarithm of both sides:
\[
\ln(1.107) = 0.017t
\]
Calculate \( \ln(1.107) \):
\[
\ln(1.107) \approx 0.102
\]
4. Now solve for \( t \):
\[
t = \frac{0.102}{0.017} \approx 6
\]
5. Since \( t \) represents the number of years after 2003, we add this to the year 2003:
\[
2003 + 6 = 2009
\]
Thus, the population of the country will be 916 million in **2009**.
1. Set up the equation:
\[
916 = 826.9 \cdot e^{0.017t}
\]
2. Divide both sides by 826.9 to isolate the exponential term:
\[
\frac{916}{826.9} = e^{0.017t}
\]
Calculating the left side:
\[
\frac{916}{826.9} \approx 1.107
\]
3. Now take the natural logarithm of both sides:
\[
\ln(1.107) = 0.017t
\]
Calculate \( \ln(1.107) \):
\[
\ln(1.107) \approx 0.102
\]
4. Now solve for \( t \):
\[
t = \frac{0.102}{0.017} \approx 6
\]
5. Since \( t \) represents the number of years after 2003, we add this to the year 2003:
\[
2003 + 6 = 2009
\]
Thus, the population of the country will be 916 million in **2009**.