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

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 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 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.

Answer 1

To analyze the given exponential function for the elephant population, we use the function:

\[ f(x) = 315 \cdot 1.075^x \]

Here:

• \( x \) represents the number of years since 1990.
• The term \( 315 \) indicates the initial population in the year 1990.
• The factor \( 1.075 \) represents the growth of the population.

1. Elephant Population in 1990: When \( x = 0 \) (which corresponds to 1990):

   \[ f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315 \]

   So, the elephant population in 1990 is 315.

2. Percentage Increase: The base of the exponent, \( 1.075 \), shows the growth factor. To find the percentage increase, we can compute:

   \[ \text{Percentage Increase} = (1.075 - 1) \times 100 = 0.075 \times 100 = 7.5% \]

Thus, the population increases by 7.5% each year.

Combining these results, we find that the correct response is:

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