Question

the elephant population of a nature preserve since 1990 is modeled by the exponential function f(x)=315·1.075^x. find the elephant population in 1990 and the percentage at which the population increases each year. (1 point)
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 0.75% each year.
the elephant population in 1990 is 339, and the population increases by 7.5% each year.

Explanation:

Step1: Find population in 1990

In 1990, \(x = 0\). Substitute \(x = 0\) into \(f(x)=315\times1.075^{x}\). Since any non - zero number to the power of 0 is 1 (\(a^{0}=1,a
eq0\)), we have \(f(0)=315\times1.075^{0}=315\times1 = 315\).

Step2: Determine growth rate

The general form of an exponential growth function is \(y = a(1 + r)^{x}\), where \(a\) is the initial amount and \(r\) is the growth rate. In the function \(f(x)=315\times1.075^{x}\), we can rewrite \(1.075\) as \(1 + 0.075\). So the growth rate \(r=0.075\), which is \(7.5\%\) when expressed as a percentage.

Answer:

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