To estimate the city's population five years from now, we need to follow the steps to understand the population decline over time as described by the given functional equation [tex]\( P = P_f \cdot a^{-0.03} \)[/tex], where [tex]\( P_f \)[/tex] is the initial population, [tex]\( a \)[/tex] is the base of the natural logarithm (approximately equal to 2.718), and [tex]\( t \)[/tex] is the time in years.
1. Identify the given values:
- Initial population ( [tex]\( P_f \)[/tex] ) is 1,000,000 (1 million).
- Base of the natural logarithm ( [tex]\( a \)[/tex] ) is approximately 2.718.
- Time ( [tex]\( t \)[/tex] ) is 5 years.
2. Determine the annual decline factor:
The annual decline factor can be represented as [tex]\( a^{-0.03} \)[/tex], where [tex]\( a = 2.718 \)[/tex].
3. Calculate the decline factor:
The decline factor is a constant value multiplied each year. Calculating this using the value provided (let's use round given, already computed):
[tex]\[
\text{decline\_factor} = 2.718^{-0.03} \approx 0.9704485521513351
\][/tex]
4. Calculate the population decline over 5 years:
To find the estimated population after 5 years, we apply the decline factor for each of the 5 years:
[tex]\[
\text{predicted\_population} = P_f \cdot (\text{decline\_factor})^5
\][/tex]
[tex]\[
\text{predicted\_population} = 1,000,000 \cdot (0.9704485521513351)^5 \approx 860,721.3628110227
\][/tex]
5. Evaluate the results:
Looking at the options provided:
- 860,721
- 223,164
- 970,448
- 1,161,816
The closest to our calculated predicted population of approximately 860,721.3628110227 is 860,721.
Therefore, the estimated population of the city five years from now is:
[tex]\[
\boxed{860,721}
\][/tex]
