To determine the population of a town after a certain number of years given an initial population and a constant growth rate, we can use the formula for exponential population growth. The formula is:
[tex]\[ P(t) = P_0 \times (1 + r)^t \][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population after [tex]\( t \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the growth rate per period (expressed as a decimal).
- [tex]\( t \)[/tex] is the number of periods (years in this case).
Given the parameters:
- Initial population, [tex]\( P_0 = 5000 \)[/tex]
- Growth rate, [tex]\( r = 0.035 \)[/tex] (which is 3.5% as a decimal)
- Number of years, [tex]\( t = 15 \)[/tex]
We substitute these values into the formula:
[tex]\[ P(15) = 5000 \times (1 + 0.035)^{15} \][/tex]
First, calculate the growth factor:
[tex]\[ 1 + 0.035 = 1.035 \][/tex]
Next, raise this growth factor to the power of 15:
[tex]\[ 1.035^{15} \approx 1.668 \][/tex]
Now, multiply this result by the initial population:
[tex]\[ P(15) = 5000 \times 1.668 \approx 8340 \][/tex]
The population after 15 years, rounded to the nearest whole number, is:
[tex]\[ \boxed{8340} \][/tex]
