Question

Birchwood Elementary 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 when reached 365, use the following function. f(n) = 240(1.15)^n After how many years did the number of students reach 365?

Answer 1

To determine how many years it took for the number of students to reach 365, we can use the provided exponential growth function:

\[ f(n) = 240(1.15)^n \]

We want to find the value of \( n \) when \( f(n) = 365 \):

\[ 240(1.15)^n = 365 \]

To solve for \( n \), we can first divide both sides by 240:

\[ (1.15)^n = \frac{365}{240} \]

Calculating the right-hand side:

\[ \frac{365}{240} \approx 1.5208333 \]

Now we have:

\[ (1.15)^n \approx 1.5208333 \]

Next, we take the logarithm of both sides to solve for \( n \):

\[ \log((1.15)^n) = \log(1.5208333) \]

Using the power rule of logarithms, we can move \( n \) in front:

\[ n \cdot \log(1.15) = \log(1.5208333) \]

Now, we can isolate \( n \):

\[ n = \frac{\log(1.5208333)}{\log(1.15)} \]

Now we can calculate the values using a calculator:

1. Calculate \( \log(1.5208333) \approx 0.1811 \)
2. Calculate \( \log(1.15) \approx 0.0614 \)
3. Now plug the values in:

\[ n \approx \frac{0.1811}{0.0614} \approx 2.95 \]

Since \( n \) represents the number of years, we can round up because it takes a full year for the number of students to reach or exceed 365. Therefore, after approximately 3 years, the number of students at Birchwood Elementary School reaches or exceeds 365.

Thus, it will take about 3 years for the student count to reach 365.