Question 10 (5 points)
An endangered species of birds has a current population of 3400. Biologists estimate that the population decreases by $8\mathrm{\%}$ 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 $P(t)=P_0\ast (1-r{)}^t$, 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 $P(t)=3400\ast (1-0.08{)}^t$.

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 $P(6)=3400\ast (1-0.08{)}^6$, 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: $\boxed{{\displaystyle 2062}}$ and $\boxed{{\displaystyle 9}}$