The elephant population of a nature preserve since 1990 is modeled by the exponential function f(x)=315⋅1.075x

Question

. 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 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.
The elephant population in 1990 is 339, 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.

Answer 1

To analyze the function \( f(x) = 315 \cdot 1.075^x \):

1. Elephant Population in 1990:

   • The variable \( x \) represents the number of years since 1990. Therefore, when \( x = 0 \) (which corresponds to the year 1990), the population is: \[ f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315. \] So, the elephant population in 1990 is 315.

2. Percentage Increase Each Year:

   • The base of the exponential function, \( 1.075 \), indicates the growth factor. The growth rate can be found by taking \( 1.075 - 1 = 0.075 \). This is equivalent to a percentage increase of: \[ 0.075 \times 100 = 7.5%. \]

Based on this analysis, the correct response is:

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