Question 15

In 2001, the National Park Service reintroduced 25 elk back into their native habitat in Great Smokey Mountains National Park.

Assume the population grew exponentially and that by 2004 there were 33 elk.

Find an exponential function of the form $f_{f}(x)=a \cdot b^{x}$ that would model this growth in the population.

Step 1 ：The problem is asking for an exponential function that models the growth of the elk population. We know that the population was 25 in 2001 and 33 in 2004. We can use these two points to find the values of a and b in the function \(f_{f}(x)=a \cdot b^{x}\).

Step 2 ：We can set up two equations using the two points (2001, 25) and (2004, 33). We can let x represent the number of years since 2001. So, for the year 2001, x = 0 and for the year 2004, x = 3.

Step 3 ：The two equations are: 1) \(f_{f}(0)=a \cdot b^{0} = 25\) 2) \(f_{f}(3)=a \cdot b^{3} = 33\)

Step 4 ：We can solve these two equations to find the values of a and b. First, we can solve the first equation for a. Since any number raised to the power of 0 is 1, we have a = 25.

Step 5 ：Then, we can substitute a = 25 into the second equation and solve for b.

Step 6 ：Finally, we can substitute the values of a and b into the function \(f_{f}(x)=a \cdot b^{x}\) to get the final function that models the growth of the elk population.

Step 7 ：Let's do these calculations. From the first equation, we find that \(a = 25\).

Step 8 ：Substituting \(a = 25\) into the second equation, we find that \(b = \frac{165^{1/3}}{5}\).

Step 9 ：Substituting these values into the function, we find that the exponential function that models the growth of the elk population is \(f_{f}(x)=25 \cdot \left(\frac{165^{1/3}}{5}\right)^{x}\).

Step 10 ：\(\boxed{f_{f}(x)=25 \cdot \left(\frac{165^{1/3}}{5}\right)^{x}}\) is the final answer.