Question 10 (5 points) An endangered species of birds has a current population of 3400. Biologists estimate that the population decreases by $8 \%$ per year. a) Write an equation that models the population of the species b) What will the population of the birds be in 6 years? c) In what year will the population be half of what it is now?

Step 1 ：We are given that the initial population of the birds is 3400 and it decreases by 8% per year. This is a typical exponential decay problem. The general form of an exponential decay function is \(P(t) = P_0 * (1 - r)^t\), where \(P_0\) is the initial population, \(r\) is the rate of decrease, and \(t\) is time in years. In this case, \(P_0 = 3400\), \(r = 0.08\), and \(t\) is the variable we'll be manipulating to answer the different parts of the question.

Step 2 ：For part a), the equation that models the population of the species is \(P(t) = 3400 * (1 - 0.08)^t\).

Step 3 ：For part b), we need to find the population of the birds in 6 years. Substituting \(t = 6\) into the equation, we get \(P(6) = 3400 * (1 - 0.08)^6\), which is approximately 2062.

Step 4 ：For part c), we need to find when the population will be half of what it is now. Setting \(P(t) = 3400/2\) and solving for \(t\), we get \(t = \log_{(1 - 0.08)}(0.5)\), which is approximately 8.31 years. So, it will be in the 9th year when the population is half of what it is now.

Step 5 ：Final Answer: \(\boxed{2062}\) and \(\boxed{9}\)