A ship carrying 1000 passengers is wrecked on a small island from which the passengers are never rescued. The natural resources of the island restrict the population to a limiting value of 5810 , to which the population gets closer and closer but which it never reaches. The population of the island after time $\mathrm{t}$, in years, is approximated by the logistic equation given below. Complete parts (a) through (c). \[ P(t)=\frac{5810}{1+4.81 e^{-0.6 t}} \] a) Find the population after 16 years. (Round to the nearest integer as needed.)

Step 1 ：The problem provides us with a logistic equation for the population of the island after time t, in years: \(P(t)=\frac{5810}{1+4.81 e^{-0.6 t}}\).

Step 2 ：We are asked to find the population after 16 years. This can be found by substituting \(t=16\) into the given logistic equation.

Step 3 ：Substituting \(t=16\) into the equation, we get \(P(16) = \frac{5810}{1+4.81 e^{-0.6 \times 16}}\).

Step 4 ：Solving this equation, we get \(P(16) = 5808.107862368711\).

Step 5 ：Rounding this to the nearest integer, we get \(P(16) = 5808\).

Step 6 ：Final Answer: The population after 16 years is approximately \(\boxed{5808}\).