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.
