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?

A.
1 year
B.
2 years
C.
3 years
D.
4 years

Answer 1

To determine after how many years the number of students at Birchwood Elementary School reached 365, we will use the function given:

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

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

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

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

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

Calculating the right side, we get:

\[ \frac{365}{240} \approx 1.52083333 \]

Now we have:

\[ (1.15)^n \approx 1.52083333 \]

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

\[ \log((1.15)^n) = \log(1.52083333) \]

Using the property of logarithms, we can bring down the exponent:

\[ n \cdot \log(1.15) = \log(1.52083333) \]

Now, we can solve for \( n \):

\[ n = \frac{\log(1.52083333)}{\log(1.15)} \]

Using a calculator, we compute:

• \( \log(1.52083333) \approx 0.1823215567939546 \)
• \( \log(1.15) \approx 0.062806139634174776 \)

Now we can calculate \( n \):

\[ n \approx \frac{0.1823215567939546}{0.062806139634174776} \approx 2.904 \]

Rounding this to the nearest whole number, we find that \( n \approx 3 \).

So, the number of years (n) that had passed when the number of students reached 365 is:

C. 3 years