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)

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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.
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 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.
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Answer 1

To determine the elephant population in 1990 and the percentage at which the population increases each year from the given function \( f(x) = 315 \cdot 1.075^x \):

1. Finding the elephant population in 1990:

   • In the model, \( x \) represents the number of years since 1990. Therefore, for the year 1990, \( x = 0 \).
   • Plugging \( x = 0 \) into the equation:

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

   So, the elephant population in 1990 is 315.

2. Finding the percentage increase in population per year:

   • The term \( 1.075 \) in the function represents the growth factor each year. This means that each year the population is multiplied by \( 1.075 \).
   • To find the percentage increase, subtract 1 from the growth factor and then convert it to a percentage:

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

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

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