Question

The elephant population of a nature preserve since 1990 is modeled by the exponential function f(x)=315⋅1.075x
. Find the elephant population in 1990 and the percentage at which the population increases each year.(1 point)
Responses

The elephant population in 1990 is 315, and the population increases by 7.5% each year.
The elephant population in 1990 is 315, and the population increases by 7.5% each year.

The elephant population in 1990 is 315, and the population increases by 1.075% each year.
The elephant population in 1990 is 315, and the population increases by 1.075% each year.

The elephant population in 1990 is 315, and the population increases by 0.75% each year.
The elephant population in 1990 is 315, and the population increases by 0.75% each year.

The elephant population in 1990 is 339, and the population increases by 7.5% each year.

Answer 1

To understand the elephant population model given by the exponential function \( f(x) = 315 \cdot 1.075^x \):

1. In this model:
- \( x \) represents the number of years since 1990.
- \( f(x) \) represents the elephant population.

2. When \( x = 0 \) (which corresponds to the year 1990):
\[
f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315
\]

So, the elephant population in 1990 is **315**.

3. The factor \( 1.075 \) indicates that the population increases by 7.5% each year. This is because:
- The annual increase rate can be calculated as \( (1.075 - 1) \times 100\% = 0.075 \times 100\% = 7.5\% \).

Putting this information together, the correct response is:

**The elephant population in 1990 is 315, and the population increases by 7.5% each year.**