To determine the predicted population in 11 years using the population decline model [tex]\( P(t) = 328,000 e^{-0.022t} \)[/tex], follow these steps:
1. Identify the given parameters:
- Initial population, [tex]\( P_0 = 328,000 \)[/tex]
- Decline rate, [tex]\( r = -0.022 \)[/tex]
- Number of years, [tex]\( t = 11 \)[/tex]
2. Substitute these values into the model:
[tex]\[
P(t) = 328,000 e^{-0.022 \times 11}
\][/tex]
3. Calculate the exponent:
[tex]\[
-0.022 \times 11 = -0.242
\][/tex]
4. Use the calculated exponent in the model:
[tex]\[
P(11) = 328,000 e^{-0.242}
\][/tex]
5. Find the value of the exponential expression [tex]\( e^{-0.242} \)[/tex]:
[tex]\[
e^{-0.242} \approx 0.784
\][/tex]
6. Multiply the initial population by this value:
[tex]\[
P(11) = 328,000 \times 0.784 \approx 257,498.4262370001
\][/tex]
7. Round the result to the nearest person:
[tex]\[
\text{Rounded population} \approx 257,498
\][/tex]
Therefore, the model predicts that the population of the inner-city area in 11 years will be approximately [tex]\( 257,498 \)[/tex] people.
