To find out how many years (n) it took for the number of students at Birchwood Elementary School to reach 365, we can use the given function:
\[ f(n) = 240(1.15)^n \]
We need to set this function equal to 365 and solve for \(n\):
\[ 240(1.15)^n = 365 \]
First, we can divide both sides by 240:
\[ (1.15)^n = \frac{365}{240} \]
Calculating the right side gives:
\[ (1.15)^n = 1.5208333 \]
Next, we apply the logarithm to both sides to solve for \(n\):
\[ \log((1.15)^n) = \log(1.5208333) \]
Using the power rule of logarithms:
\[ n \log(1.15) = \log(1.5208333) \]
Now, we can solve for \(n\):
\[ n = \frac{\log(1.5208333)}{\log(1.15)} \]
Calculating the logarithms using a calculator:
- \( \log(1.5208333) \approx 0.1827 \)
- \( \log(1.15) \approx 0.0672 \)
Now we plug in the values:
\[ n \approx \frac{0.1827}{0.0672} \approx 2.72 \]
Since \(n\) must be a whole number, we round to the nearest whole number. Thus, the number of years it took for the number of students to reach at least 365 is:
\[ n = 3 \]
So, it took approximately 3 years for the student population to reach 365.