Question

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) = ?

Answer 1

9514 1404 393

Answer:

n(t) = 20,000/(1 +99e^(-0.7033t))

Explanation:

One way to write the logistic function is ...

n(t) = L/(1 +a·e^(-kt))

where L is the maximum value and parameters 'a' and k depend on boundary conditions.

Here, we have n(∞) = L = 20,000 and n(0) = L/(1+a) = 200. We also have n(1) = L/(1+a·e^-k) = 400.

Solving for 'a', we get ...

n(0) = 20000/(1+a) = 200

20000/200 -1 = a = 99

Solving for k, we get ...

n(1) = 20000/(1 +99e^-k) = 400

20000/400 -1 = 99e^-k = 49

e^-k = 49/99

k = -ln(49/99) ≈ 0.7033

So, the desired function is ...

n(t) = 20000/(1 +99e^(-0.7033t))