Question

When Birchwood Elementary School opened for the first year, there were 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 equation.

240
(
1.15
)
n
=
365

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 solve for the number of years \( n \) when the number of students reached 365, we can use the provided equation:

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

First, we divide both sides by 240 to isolate the exponential term:

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

Calculating \( \frac{365}{240} \):

\[
\frac{365}{240} \approx 1.520833
\]

Now we have:

\[
(1.15)^n \approx 1.520833
\]

Next, we need to solve for \( n \) by taking the logarithm of both sides. We can use any logarithm, but we'll use the natural logarithm (ln) for this example:

\[
\ln((1.15)^n) = \ln(1.520833)
\]

Using the power rule of logarithms, we have:

\[
n \cdot \ln(1.15) = \ln(1.520833)
\]

Now, we can calculate \( \ln(1.15) \) and \( \ln(1.520833) \):

1. Calculate \( \ln(1.15) \):

\[
\ln(1.15) \approx 0.1397619
\]

2. Calculate \( \ln(1.520833) \):

\[
\ln(1.520833) \approx 0.419555
\]

Now, substituting these values in, we have:

\[
n \cdot 0.1397619 = 0.419555
\]

To solve for \( n \):

\[
n = \frac{0.419555}{0.1397619} \approx 3.00
\]

Therefore, after about **3 years**, the number of students reached 365.

The answer is:

**C. 3 years**