When Birchwood Elementrary School opened for the first year, there 240 students. During each of the following 10 years, the number of students increased exponentially. To determine the number of years (n) that had passed when the number of students reached 365, use the following function.

Question

f (n) = 240(1.15)^n

After how many years did the number of students reach 365?

Answer 1

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:

1. \( \log(1.5208333) \approx 0.1827 \)
2. \( \log(1.15) \approx 0.0672 \)

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.