To determine the population of the inner-city area in 9 years, we follow the steps given in the exponential decay model [tex]\(P(t) = 370,000 e^{-0.011t}\)[/tex], where [tex]\(t\)[/tex] represents time in years.
Step-by-Step Solution:
1. Identify the Parameters:
- Initial population, [tex]\(P_0\)[/tex] (present population) = 370,000
- Decay rate, [tex]\(r\)[/tex] = -0.011
- Time, [tex]\(t\)[/tex] = 9 years
2. Substitute the Values:
Substitute [tex]\(P_0 = 370,000\)[/tex], [tex]\(r = -0.011\)[/tex] and [tex]\(t = 9\)[/tex] into the model [tex]\(P(t) = 370,000 e^{-0.011 \times 9}\)[/tex].
3. Calculate the Exponent:
First, calculate the exponent:
[tex]\[
-0.011 \times 9 = -0.099
\][/tex]
4. Compute the Exponential Term:
Next, calculate the exponential term [tex]\(e^{-0.099}\)[/tex]:
[tex]\[
e^{-0.099} \approx 0.905744
\][/tex]
5. Multiply by the Initial Population:
Multiply this result by the initial population to get the future population:
[tex]\[
P(9) = 370,000 \times 0.905744 \approx 335,124.80196871294
\][/tex]
6. Round to the Nearest Person:
Finally, round the result to the nearest person:
[tex]\[
P(9) \approx 335,125
\][/tex]
Conclusion:
The model predicts that the population of the inner-city area in 9 years will be approximately 335,125 people.
