To estimate the population in 2004 using the exponential growth formula, let's go through the steps to calculate it.
1. Population Data:
- Initial population ([tex]\(P_0\)[/tex]) in 1993: [tex]\(P_{1993} = 83\)[/tex] million
- Population in 2001: [tex]\(P_{2001} = 87\)[/tex] million
- Years:
- [tex]\(t_1 = 1993\)[/tex]
- [tex]\(t_2 = 2001\)[/tex]
- We need to estimate the population for [tex]\(t_3 = 2004\)[/tex]
2. Growth Rate (k):
We need to find the growth rate [tex]\(k\)[/tex] using the population data.
The exponential growth formula is:
[tex]\[
P(t) = P_0 \cdot e^{k \cdot (t - t_0)}
\][/tex]
For 2001:
[tex]\[
P_{2001} = P_{1993} \cdot e^{k \cdot (t_2 - t_1)}
\][/tex]
Substitute the known values:
[tex]\[
87 = 83 \cdot e^{k \cdot (2001 - 1993)}
\][/tex]
Simplify the exponent:
[tex]\[
87 = 83 \cdot e^{k \cdot 8}
\][/tex]
3. Solve for [tex]\(k\)[/tex]:
[tex]\[
\frac{87}{83} = e^{8k}
\][/tex]
Take the natural logarithm on both sides to isolate [tex]\(k\)[/tex]:
[tex]\[
\ln \left(\frac{87}{83}\right) = 8k
\][/tex]
[tex]\[
k = \frac{\ln \left(\frac{87}{83}\right)}{8}
\][/tex]
Through this calculation, we find:
[tex]\[
k \approx 0.005883
\][/tex]
4. Estimate Population in 2004:
Use the growth rate [tex]\(k\)[/tex] to estimate the population in 2004.
[tex]\[
P_{2004} = P_{1993} \cdot e^{k \cdot (2004 - 1993)}
\][/tex]
Substitute the known values:
[tex]\[
P_{2004} = 83 \cdot e^{0.005883 \cdot 11}
\][/tex]
[tex]\[
P_{2004} \approx 83 \cdot e^{0.064713}
\][/tex]
Solving this, we get:
[tex]\[
P_{2004} \approx 88.549
\][/tex]
5. Round to the Nearest Million:
The population estimated to the nearest million is:
[tex]\[
P_{2004} \approx 89 \text{ million}
\][/tex]
Thus, the estimated population in 2004 is [tex]\(\boxed{89}\)[/tex] million.
