Chapter 7.5, Problem 32E

a.

To determine

To show that the population function is a solution of a logistic differential equation and identify value of $k$ and the carrying capacity.

Answer to Problem 32E

The given population function is a solution of a logistic differential equation and the value of $k=0.005$ and $M=200$ .

Explanation of Solution

Given:

The number of students infected by measles in a certain school is given by the formula

  $P\left(t\right)=\frac{200}{1+e^{5.3-t}}$

Where $t$ is the number of days after students are first exposed to an infected student.

Calculation:

Since, the solution o the general logistic differential equation

  $\frac{dP}{dt}=kP\left(M-P\right)$

is given by

  $P=\frac{M}{1+A{e}^{-\left(Mk\right)t}}$

where $M$ is carrying capacity and $k$ is growth constant,

therefore, given population function is, $P\left(t\right)=\frac{200}{1+e^{5.3-t}}$ rewriting it in form $P=\frac{M}{1+A{e}^{-\left(Mk\right)t}}$

  $\begin{array}{l}P\left(t\right)=\frac{200}{1+e^{5.3-t}}\\ P\left(t\right)=\frac{200}{1+e^{5.3}{e}^{-t}}\\ \therefore A=e^{5.3},M=200\\ -t=-\left(Mk\right)t\\ -t=-\left(200k\right)t\\ 1=200k\\ k=0.005\end{array}$

Therefore, the given population function is a solution of a logistic differential equation and the value of $k=0.005$ and $M=200$ .

b.

To determine

To estimate $P\left(0\right)$

Answer to Problem 32E

The value of $P\left(0\right)=1$ .

Explanation of Solution

Given:

The number of students infected by measles in a certain school is given by the formula

  $P\left(t\right)=\frac{200}{1+e^{5.3-t}}$

Where $t$ is the number of days after students are first exposed to an infected student.

Calculation:

Since ,

  $\begin{array}{l}P\left(t\right)=\frac{200}{1+e^{5.3-t}}\\ P\left(0\right)=\frac{200}{1+e^{5.3-0}}\approx 1\end{array}$

Hence, it can be concluded that when $t=0$ ,only $1$ student is infected by measles.