To determine the population of a town after 5 years, given an initial population of 10,000 and an annual growth rate of 5%, we use the concept of exponential growth. Here's a step-by-step explanation of the solution:
1. Identify the initial population and growth rate:
- Initial population (\(\( P_0 \)\)): 10,000
- Annual growth rate (\(\( r \)\)): 5% or 0.05 (as a decimal)
2. Determine the number of years the population grows:
- Number of years (\(\( t \)\)): 5
3. Formula for exponential growth:
The population after \(\( t \)\) years can be calculated using the compound interest formula:
\(\[
P(t) = P_0 \times (1 + r)^t
\]\)
where:
- \(\( P(t) \)\) is the population after \(\( t \)\) years
- \(\( P_0 \)\) is the initial population
- \(\( r \)\) is the growth rate
- \(\( t \)\) is the number of years
4. Substitute the given values into the formula:
\(\[
P(5) = 10000 \times (1 + 0.05)^5
\]\)
\(\[
P(5) = 10000 \times (1.05)^5
\]\)
5. Calculate \(\( (1.05)^5 \)\):
\(\[
(1.05)^5 = 1.2762815625
\]\)
6. Multiply the initial population by the growth factor:
\(\[
P(5) = 10000 \times 1.2762815625
\]\)
\(\[
P(5) = 12762.815625
\]\)
7. Round the population to the nearest whole number:
\(\[
P(5) \approx 12763
\]\)
Thus, the population of the town after 5 years, rounded to the nearest whole number, will be 12,763.
