IDNLearn.com is designed to help you find reliable answers to any question you have. Ask anything and receive well-informed answers from our community of experienced professionals.
Sagot :
To estimate the population of a country in 2009 using the exponential growth formula [tex]\( P = A e^{kt} \)[/tex], follow these steps:
1. Identify the given data:
- Initial population in 1992 ([tex]\(P_{initial}\)[/tex]): 39 million
- Population in 1997 ([tex]\(P_{1997}\)[/tex]): 42 million
- Time difference from 1992 to 1997 ([tex]\(t_{1997}\)[/tex]): 1997 - 1992 = 5 years
- Time difference from 1992 to 2009 ([tex]\(t_{2009}\)[/tex]): 2009 - 1992 = 17 years
2. Set up the equation for the population in 1997:
[tex]\[ P_{1997} = P_{initial} \cdot e^{k \cdot t_{1997}} \][/tex]
Plug in the known values:
[tex]\[ 42 = 39 \cdot e^{k \cdot 5} \][/tex]
3. Solve for the growth rate [tex]\(k\)[/tex]:
- First, isolate [tex]\(e^{k \cdot 5}\)[/tex]:
[tex]\[ \frac{42}{39} = e^{5k} \][/tex]
[tex]\[ 1.0769 = e^{5k} \][/tex]
- Take the natural logarithm of both sides to solve for [tex]\(k\)[/tex]:
[tex]\[ \ln(1.0769) = 5k \][/tex]
[tex]\[ k = \frac{\ln(1.0769)}{5} \][/tex]
- Compute [tex]\( \ln(1.0769) \approx 0.0741 \)[/tex] (rounded to four decimal places):
[tex]\[ k \approx \frac{0.0741}{5} \approx 0.0148 \][/tex]
4. Use the value of [tex]\(k\)[/tex] to estimate the population in 2009:
- Set up the equation for the population in 2009:
[tex]\[ P_{2009} = P_{initial} \cdot e^{k \cdot t_{2009}} \][/tex]
Plug in the values:
[tex]\[ P_{2009} = 39 \cdot e^{0.0148 \cdot 17} \][/tex]
- Compute the exponent:
[tex]\[ 0.0148 \cdot 17 \approx 0.2516 \][/tex]
- Compute [tex]\(e^{0.2516} \approx 1.285\)[/tex]:
[tex]\[ P_{2009} = 39 \cdot 1.285 \approx 50.1756 \][/tex]
5. Round the estimated population to the nearest million:
[tex]\[ P_{2009} \approx 50 \][/tex]
Therefore, the estimated population of the country in 2009 is approximately 50 million.
1. Identify the given data:
- Initial population in 1992 ([tex]\(P_{initial}\)[/tex]): 39 million
- Population in 1997 ([tex]\(P_{1997}\)[/tex]): 42 million
- Time difference from 1992 to 1997 ([tex]\(t_{1997}\)[/tex]): 1997 - 1992 = 5 years
- Time difference from 1992 to 2009 ([tex]\(t_{2009}\)[/tex]): 2009 - 1992 = 17 years
2. Set up the equation for the population in 1997:
[tex]\[ P_{1997} = P_{initial} \cdot e^{k \cdot t_{1997}} \][/tex]
Plug in the known values:
[tex]\[ 42 = 39 \cdot e^{k \cdot 5} \][/tex]
3. Solve for the growth rate [tex]\(k\)[/tex]:
- First, isolate [tex]\(e^{k \cdot 5}\)[/tex]:
[tex]\[ \frac{42}{39} = e^{5k} \][/tex]
[tex]\[ 1.0769 = e^{5k} \][/tex]
- Take the natural logarithm of both sides to solve for [tex]\(k\)[/tex]:
[tex]\[ \ln(1.0769) = 5k \][/tex]
[tex]\[ k = \frac{\ln(1.0769)}{5} \][/tex]
- Compute [tex]\( \ln(1.0769) \approx 0.0741 \)[/tex] (rounded to four decimal places):
[tex]\[ k \approx \frac{0.0741}{5} \approx 0.0148 \][/tex]
4. Use the value of [tex]\(k\)[/tex] to estimate the population in 2009:
- Set up the equation for the population in 2009:
[tex]\[ P_{2009} = P_{initial} \cdot e^{k \cdot t_{2009}} \][/tex]
Plug in the values:
[tex]\[ P_{2009} = 39 \cdot e^{0.0148 \cdot 17} \][/tex]
- Compute the exponent:
[tex]\[ 0.0148 \cdot 17 \approx 0.2516 \][/tex]
- Compute [tex]\(e^{0.2516} \approx 1.285\)[/tex]:
[tex]\[ P_{2009} = 39 \cdot 1.285 \approx 50.1756 \][/tex]
5. Round the estimated population to the nearest million:
[tex]\[ P_{2009} \approx 50 \][/tex]
Therefore, the estimated population of the country in 2009 is approximately 50 million.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.