Question

In Exercises 31 and \(32,\) a population function is given. (a) Show that the function is a solution of a logistic differential equation. Identify \(k\) and the carrying capacity. (b) Writing to Learn Estimate \(P(0)\) . Explain its meaning in the context of the problem. Spread of Measles The number of students infected by measles in a certain school is given by the formula \(P(t)=\frac{200}{1+e^{5.3-t}}\) where \(t\) is the number of days after students are first exposed to an infected student.

Short answer

The given population function can indeed be put into logistic form with a carrying capacity \(K = 200\) and growth rate \(k = 1\). The quantity \(P(0)=\frac{200}{1+e^{5.3}}\) gives the initial infected population at the start of the infection spread.

Step-by-step solution

Verify the Logistic Equation Form

The general form of a logistic function is given by \(P(t) = \frac{K}{1+ C e^{(-kt)}}\), where K represents the carrying capacity, and k is the growth rate. We will arrange the given function \(P(t)=\frac{200}{1+e^{(5.3-t)}}\) to this format and prove it to be a logistic function.

Identify the Parameters

Comparing the given function with the generic form, it can be seen that the carrying capacity \(K = 200\). The growth rate can be found as \(k = 1\), as it is the coefficient of t in the exponent.

Evaluate for P(0)

If we plug t = 0 into the given function, it gives the initial population who are infected by measles, that is, \(P(0)=\frac{200}{1+e^{5.3}}\).

Interpret the meaning of P(0)

Now, \(P(0)\) stands for the infected population just when the students are first exposed to an infected student. In the context of this problem, it is crucial as it provides the initial state or the starting point of the disease spread.