The number N(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. Initially, N(0) = 200, and it is observed that N(1) = 400. Solve for N(t) if it is predicted that the limiting number of people in the community who will see the advertisement is 20,000. (Round all coefficients to four decimal places.) N(t) =

The solution to the corresponding initial-value problem is given by

.

Step-by-step explanation:

Differential equations can be used to represent the size of a population as it varies over time.

The logistics equation is a differential equation that models population growth and is given by

where is the carrying capacity, it is the maximum population that the environment can support.

With constants relabeled, the nonlinear equation is the same as

The logistic differential equation is an autonomous differential equation, so we can use separation of variables to find the general solution.

Consider the logistic differential equation subject to an initial population of with carrying capacity and growth rate . The solution to the corresponding initial-value problem is given by

or

From here we get:

as

From the information given:

So, the logistic equation is

Since the limiting number is 20,000, we get that

Next, we divide by in the numerator and the denominator.

To find the value of the constant we use the initial condition .

The solution to the corresponding initial-value problem is given by

20

step-by-step explanation:

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step-by-step explanation: