Questions: 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 5740, 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) = 5740 / (1 + 4.74 e^(-0.4 t)) a) Find the population after 15 years. (Round to the nearest integer as needed.)

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 5740, 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) = 5740 / (1 + 4.74 e^(-0.4 t)) a) Find the population after 15 years. (Round to the nearest integer as needed.)
Transcript text: 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 5740 , 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 ( $c$ ). \[ P(t)=\frac{5740}{1+4.74 e^{-0.4 t}} \] a) Find the population after 15 years. $\square$ (Round to the nearest integer as needed.)
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Solution

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Solution Steps

Step 1: Identify the Parameters

The given parameters are: L (carrying capacity) = 5740, A (related to the initial population size) = 4.74, k (relative growth rate) = 0.4, and t (time) = 15.

Step 2: Substitute the Parameters into the Logistic Equation

Substituting the given parameters into the logistic growth model equation, we get: \[P(t) = \frac{5740}{1 + 4.74e^{-0.4t}}\].

Step 3: Solve for P(t)

Substituting t = 15 into the equation, we find the population at time t to be \[P(15) = 5673\].

Step 4: Find the Rate of Change of the Population at Time t

The rate of change of the population at time t, denoted as \(P'(15)\), is calculated by differentiating \(P(t)\) with respect to \(t\) and substituting the specific value of \(t\). This gives us \[P'(15) = 26\].

Final Answer:

The population at time \(t = 15\) is approximately 5673, and the rate of change of the population at this time is approximately 26.

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