To solve this problem, we need to determine the student population in Seattle 7 years from now, given an annual decrease rate of 9%.
Step-by-Step Solution:
1. Understanding the Decrease Rate:
- The population decreases by 9% annually. This means if the current population is [tex]\( P_0 \)[/tex], the population after 1 year will be 91% of [tex]\( P_0 \)[/tex] (100% - 9%).
2. Exponential Model:
- The general formula for population decrease is given by the exponential decay model:
[tex]\[
P(x) = P_0 \cdot (1 - r)^x
\][/tex]
where:
- [tex]\( P(x) \)[/tex] is the population after [tex]\( x \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population (2400).
- [tex]\( r \)[/tex] is the annual decrease rate (0.09).
- [tex]\( x \)[/tex] is the number of years.
Plugging the values into the formula:
[tex]\[
P(x) = 2400 \left(1 - 0.09 \right)^x = 2400 \left(0.91 \right)^x
\][/tex]
Thus, the exponential model for the population decrease over [tex]\( x \)[/tex] years is:
[tex]\[
P(x) = 2400 \cdot (0.91)^x
\][/tex]
3. Calculating the Population in 7 Years:
- Substitute [tex]\( x = 7 \)[/tex] into the model:
[tex]\[
P(7) = 2400 \cdot (0.91)^7
\][/tex]
4. Population Calculation Over Each Year:
- Calculate the population year by year to understand the trend:
[tex]\[
\begin{align*}
P(1) &= 2400 \cdot 0.91 = 2184.0 \\
P(2) &= 2184.0 \cdot 0.91 = 1987.44 \\
P(3) &= 1987.44 \cdot 0.91 = 1808.5704 \\
P(4) &= 1808.5704 \cdot 0.91 = 1645.799064 \\
P(5) &= 1645.799064 \cdot 0.91 = 1497.67714824 \\
P(6) &= 1497.67714824 \cdot 0.91 = 1362.88620490 \\
P(7) &= 1362.88620490 \cdot 0.91 = 1240.22644646 \\
\end{align*}
\][/tex]
5. Final Answer:
- After 7 years, the student population in Seattle will be approximately 1240. Round to the nearest whole number:
[tex]\[
\boxed{1240}
\][/tex]
So, the population in 7 years will be 1240 students.
