To determine the population of the city seven years from now, let's break down the problem step by step:
1. Current Population:
The city currently has a population of 33,000 residents.
2. Growth Rate:
The city is adding new residents at a steady rate of 1,800 residents per year.
3. Time Period:
We need to calculate the population after 7 years.
4. Resident Proportion Function:
The proportion of residents who remain after [tex]\( t \)[/tex] years is given by [tex]\( S(t) = \frac{1}{t+1} \)[/tex]. Here, [tex]\( t = 7 \)[/tex] years, so we compute [tex]\( S(t) \)[/tex] as follows:
[tex]\[
S(7) = \frac{1}{7+1} = \frac{1}{8} = 0.125
\][/tex]
5. Number of New Residents in 7 Years:
Over a period of 7 years, the number of new residents added is:
[tex]\[
\text{New Residents} = \text{Growth Rate} \times t = 1800 \, \text{residents/year} \times 7 \, \text{years} = 12600 \, \text{residents}
\][/tex]
6. Remaining Population from Original Residents:
After 7 years, the proportion of the original population that remains is [tex]\( 0.125 \)[/tex], so the remaining number of original residents is:
[tex]\[
\text{Remaining Original Residents} = \text{Current Population} \times S(7) = 33000 \times 0.125 = 4125 \, \text{residents}
\][/tex]
7. Future Population Calculation:
The future population of the city includes the remaining original residents and the new residents who have moved in over the 7 years. Therefore, the total future population is:
[tex]\[
\text{Future Population} = \text{Remaining Original Residents} + \text{New Residents}
\][/tex]
[tex]\[
\text{Future Population} = 4125 \, \text{residents} + 12600 \, \text{residents} = 16725 \, \text{residents}
\][/tex]
Thus, the population of the city seven years from now will be 16,725 residents.
