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

Question

𝑓
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315
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1.075
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. 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 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 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.

Answer 1

To analyze the provided exponential function \( f(x) = 315 \cdot 1.075^x \), we can break it down as follows:

1. Finding the elephant population in 1990:
   In this model, \( x \) represents the number of years since 1990. Therefore, when \( x = 0 \) (which corresponds to the year 1990), we calculate:

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

   So, the elephant population in 1990 is 315.

2. Finding the percentage increase in population each year:
   The base of the exponent in the function, \( 1.075 \), indicates that the population increases by a factor of 1.075 each year. To find the percentage increase, we subtract 1 and convert it to a percentage:

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

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

Hence, the correct response is:

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