Question

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 5790, to which the population gets closer and closer but which it never reaches. the population of the island after time t, in years, is approximated by the logistic equation given below. complete parts (a) through (c)
p(t)=\frac{5790}{1 + 4.79e^{-0.6t}}
a) find the population after 12 years
(round to the nearest integer as needed.)

Explanation:

Step1: Substitute $t = 12$ into the formula

Substitute $t = 12$ into $P(t)=\frac{5790}{1 + 4.79e^{-0.6t}}$. So we get $P(12)=\frac{5790}{1 + 4.79e^{-0.6\times12}}$.

Step2: Calculate the exponent part

First, calculate $-0.6\times12=-7.2$. Then $e^{-7.2}\approx0.000747$.

Step3: Calculate the denominator part

Next, calculate $4.79\times e^{-7.2}=4.79\times0.000747\approx0.0036$. Then $1 + 4.79e^{-7.2}=1+ 0.0036 = 1.0036$.

Step4: Calculate the population value

Finally, $P(12)=\frac{5790}{1.0036}\approx5770$.

Answer:

5770