To determine the population of the town in the year 2035 using the given model [tex]\( P = -20n^2 + 650n + 2550 \)[/tex], follow these steps:
1. Identify [tex]\(n\)[/tex]:
- [tex]\( n \)[/tex] represents the number of years since 2020.
- To find [tex]\( n \)[/tex] for the year 2035, we calculate the difference between 2035 and 2020:
[tex]\[
n = 2035 - 2020 = 15
\][/tex]
2. Substitute [tex]\(n\)[/tex] into the equation:
- We substitute [tex]\( n = 15 \)[/tex] into the population model [tex]\( P = -20n^2 + 650n + 2550 \)[/tex].
3. Evaluate the expression:
- First, calculate the term with [tex]\( n^2 \)[/tex]:
[tex]\[
-20 \cdot (15)^2 = -20 \cdot 225 = -4500
\][/tex]
- Next, calculate the term with [tex]\( n \)[/tex]:
[tex]\[
650 \cdot 15 = 9750
\][/tex]
- Finally, add the constant term:
[tex]\[
2550
\][/tex]
4. Combine all the terms:
- Add the results from each step:
[tex]\[
P = -4500 + 9750 + 2550
\][/tex]
[tex]\[
P = 7800
\][/tex]
Therefore, the population of the town in the year 2035 is [tex]\( 7800 \)[/tex].
