To determine how many years it took for the student population at Birchwood Elementary School to grow from 240 to 365, we need to consider the exponential growth model.
Assuming exponential growth is described with the formula:
\[ N(t) = N_0 \cdot e^{kt} \]
where:
- \( N(t) \) is the number of students at time \( t \),
- \( N_0 \) is the initial number of students (240 in this case),
- \( k \) is a constant that represents the growth rate,
- \( t \) is the time in years.
However, to simplify, we can use an alternative approach given that we have a specific number of years (10) to track the growth towards specific values.
To find how long it takes to reach 365 students under exponential growth, we can also use a direct simple calculation if we can assume a certain yearly growth factor.
Let's first calculate the approximate growth factor needed in each year:
\[ \text{Let the number of students grow by a factor of } r \text{ each year.} \]
The equation can be modeled as follows over \( n \) years: \[ 240 \times r^n = 365 \]
After determining \( r \) or through approximations, we can substitute different years (1 to 4) in:
-
After 1 year: \[ 240 \cdot r = 365 \Rightarrow r = \frac{365}{240} \approx 1.52 \text{ (too high)} \]
-
After 2 years: \[ 240 \cdot r^2 = 365 \Rightarrow r^2 = \frac{365}{240} \approx 1.52 \Rightarrow r \approx \sqrt{1.52} \approx 1.23 \text{ (possible)} \]
-
After 3 years: \[ 240 \cdot r^3 = 365 \]
-
After 4 years: \[ 240 \cdot r^4 = 365 \Rightarrow r^4 = \frac{365}{240} \text{ gives a larger growth factor, probably too high.} \]
Simplicity suggests the first calculations imply that it took either 2 or 3 years to grow from 240 to 365.
Using more of a software-assisted or calculated logarithmic discovery step can finalize: \[ n = \frac{\log(365/240)}{\log(r)} \]
The values we've found suggest that the growth likely can be approximated as:
- Answer: B. 2 years.
This is estimating based on typical scenarios for similar growth and scaling. For more exact results, experimental data regarding exactly r or historical analysis would bolster accuracy, else simplifying yields meaningful conclusions towards the representative option.