The elephant population of a nature preserve since 1990 is modeled by the exponential function

Question

f
(
x
)
=
315
⋅
1.075
x
. 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 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 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 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 exponential function modeling the elephant population:

The function given is: \[ f(x) = 315 \cdot 1.075^x \]

In this function, \( x \) represents the number of years since 1990.

1. Finding the initial population in 1990: 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.

2. Finding the percentage increase each year: The growth factor is \( 1.075 \). The percentage increase can be found by subtracting 1 from the growth factor and converting it to a percentage: \[ \text{Percentage increase} = (1.075 - 1) \cdot 100% = 0.075 \cdot 100% = 7.5% \] Thus, the elephant population increases by 7.5% each year.

Given these findings, the correct response is: The elephant population in 1990 is 315, and the population increases by 7.5% each year.