Question
A town has a population of 13000 and grows at \( 5 \% \) every year. What will be the population after 11 years, to the nearest whole number?
Ask by Norton Mitchell. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
After 11 years, the population will be 22,234.
Solution
To calculate the population of the town after 11 years with an annual growth rate of \( 5\% \), you can use the formula for compound growth:
\[
P = P_0 \times (1 + r)^n
\]
Where:
- \( P_0 \) is the initial population (13,000)
- \( r \) is the growth rate (5\% or 0.05)
- \( n \) is the number of years (11)
Plugging in the values:
\[
P = 13000 \times (1.05)^{11}
\]
First, calculate \( (1.05)^{11} \):
\[
1.05^{11} \approx 1.71034
\]
Now, multiply this by the initial population:
\[
P \approx 13000 \times 1.71034 \approx 22234.42
\]
Rounding to the nearest whole number, the population after 11 years will be **22,234**.
**Answer:**
After 11 years, the population will be 22 234.
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Simplify this solution Extra Insights
To find the population after 11 years, you can use the formula for exponential growth: \[ P = P_0 (1 + r)^t \] where: - \(P\) is the future population, - \(P_0\) is the initial population (13,000), - \(r\) is the growth rate (0.05), and - \(t\) is the time in years (11). Plugging in the values: \[ P = 13000 (1 + 0.05)^{11} \] \[ P = 13000 (1.05)^{11} \] Calculating \( (1.05)^{11} \) gives approximately 1.7137. Now, multiply: \[ P \approx 13000 \times 1.7137 \approx 22278.1 \] Rounding to the nearest whole number, the population after 11 years will be approximately 22,278.
