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