Question
The elephant population of a nature preserve since 1990 is modeled by the exponential function f (x) = 315 • 1.075*
population increases each year. (1 point)
A. The elephant population in 1990 is 339 and the population increases by 7.5% each year
B. The elephant population in 1990 is 315 and the population increases by 1.075% each year
C. The elephant population in 1990 is 315 and the population increases by 7.5% each year.
D. The animal population in 1990 is 315 and the population increases by 0.75% each year
population increases each year. (1 point)
A. The elephant population in 1990 is 339 and the population increases by 7.5% each year
B. The elephant population in 1990 is 315 and the population increases by 1.075% each year
C. The elephant population in 1990 is 315 and the population increases by 7.5% each year.
D. The animal population in 1990 is 315 and the population increases by 0.75% each year
Answers
GPT-4o mini
The function provided is \( f(x) = 315 \cdot 1.075^x \).
From the function, we can analyze the components:
1. **Initial Population**: The value at \( x = 0 \) (which corresponds to the year 1990) is \( f(0) = 315 \cdot 1.075^0 = 315 \). This tells us that the elephant population in 1990 is 315.
2. **Growth Rate**: The base of the exponential function (1.075) represents the growth factor. To find the percentage increase, we subtract 1 from this value and convert it to a percentage:
\[
1.075 - 1 = 0.075 \quad \text{(which is 7.5% when converted to a percentage)}
\]
So, the elephant population increases by 7.5% each year.
Based on this analysis, the correct option is:
**C. The elephant population in 1990 is 315 and the population increases by 7.5% each year.**
From the function, we can analyze the components:
1. **Initial Population**: The value at \( x = 0 \) (which corresponds to the year 1990) is \( f(0) = 315 \cdot 1.075^0 = 315 \). This tells us that the elephant population in 1990 is 315.
2. **Growth Rate**: The base of the exponential function (1.075) represents the growth factor. To find the percentage increase, we subtract 1 from this value and convert it to a percentage:
\[
1.075 - 1 = 0.075 \quad \text{(which is 7.5% when converted to a percentage)}
\]
So, the elephant population increases by 7.5% each year.
Based on this analysis, the correct option is:
**C. The elephant population in 1990 is 315 and the population increases by 7.5% each year.**