In an effort to control vegetation overgrowth, 81 rabbits are released in an isolated area free of predators.
Step 1 of 2 : Find the equation of the exponential function if the rabbit population has increased to 162 after 1 year.

Answer 2 Points
\[
p(t)=1
\]

Step 1 ：The problem is asking for an exponential function that models the growth of the rabbit population over time. The general form of an exponential function is \(p(t) = a \cdot b^{t}\), where \(p(t)\) is the population at time \(t\), \(a\) is the initial population, \(b\) is the growth factor, and \(t\) is the time in years.

Step 2 ：We know that the initial population (\(a\)) is 81 rabbits, and after 1 year (\(t=1\)), the population has doubled to 162 rabbits. This means that the growth factor (\(b\)) is 2.

Step 3 ：So, the equation of the exponential function is \(p(t) = 81 \cdot 2^{t}\).

Step 4 ：Let's check if this function gives the correct population after 1 year. The function should return 162, which is the population of rabbits after 1 year.

Step 5 ：The function returned 162, which is the correct population of rabbits after 1 year. Therefore, the exponential function \(p(t) = 81 \cdot 2^{t}\) is correct.

Step 6 ：\(\boxed{p(t) = 81 \cdot 2^{t}}\) is the equation of the exponential function that models the growth of the rabbit population over time.